Another suggestion for fellow native speakers: if someone asks you "Can you repeat that?", don't rephrase what you said, just repeat the exact same words in the same order.

Lots of people are perfectly capable of understanding what you said, you just said it too fast!

I'm a native English speaker who lived in a country with a different language for most of my adult life. There is a huge difference between someone who consciously adapts their speaking and a random person on the street. The difference can be as stark as me understanding 95-99% of what is said and me understanding 20-50% of what is said.

Adapting one's speaking for a non-native speaker is a distinct skill, and (my guess is that) most native English speakers have never even considered it. While things like accent are harder to be aware of and to control, word choice and speed are easy to adapt. For anyone looking for tips, here is what helped me:

• Speaking at normal pace or at 90-95% of a normal pace was very helpful, as opposed to speaking at an I'm-so-excited-about-this pace.
• Using words that are common (such as "location" or "building" rather than "venue") were a big help for me. Often when I didn't understand something, it was simply because the speaker used a word I didn't know, but I knew a more common/simple word for the same thing (such as using the word "residence" or "domicile" instead of using the world "house" or "home").
• Prefacing or giving a subject line. Instead of starting a new topic directly, first say "I'd like to talk about TOPIC."

It can be easy to accidently appear patronizing if you slow down too much or over-enunciate, so try to be aware of the other person and of your social dynamics with the person.

Here are some things that anecdotally helped me as a non-native speaker in EA (although I consider myself relatively fluent), followed by some of my thoughts for native speakers (after the numbered list).

1. You mention:

I often just nod as if I understand what is being said because I fear that by asking "Can you repeat that, please?" I will be perceived as stupid and slow in a community that values time, effectiveness, high-value actions, and reason. I understand that this is mainly my own issue, and I am working on improving my language skills, but I think that something might be done on the other side too.

I strongly feel (90%) that asking for clarifications/explanations usually makes someone look smarter and more interested, and we don't do this enough. It seems to me that people consider it a strong signal that you're actually listening, interested in what the other person is talking about, and you're actually trying to understand what they're saying. Also, at EAGx sometimes even proficient English speakers didn't know what a word meant (examples:  MC/"master of ceremonies" and "marquee", apparently they're British words), asking for clarifications on those gives useful feedback to speakers on what terms are more common.

Anecdote: the only such case where I think people might have thought less of me was when I asked what a ToC was, and after that asked what a Theory of Change was, in a course where the material that week was about Theories of Change, and I hadn't read any of it. Ooof. I still think that I counterfactually looked less stupid by asking, since I was more able to follow the discussion.

I know this doesn't help much ("be less shy" is not great advice) but I think in EA asking for clarifications usually makes you look smarter, at least counterfactually.

2. Try to have talks in quiet places, I'm still surprised by how much easier it is for me to hold a conversation in English in a quiet place compared to a crowded place. I feel barely any difference in Italian. I think spending 5 minutes moving to a quieter table, a quieter room, or outside is worth it for me. Even for a 30-minute conversation. (To "abuse IT jargon every two sentences to express easy concepts": I find the SNR counterintuitively important when speaking in English)

3. Focus even more than you normally would on the person you're speaking with (I think this is good advice in general). I found that making a conscious effort to ensure that my interlocutor has 110% of my attention helped immensely. (Related, I expect all usual conversation tips to be even more useful in a second language).

4. Alcohol makes both speaking and understanding much harder. The effect for me is like 10x higher in English than in Italian.

5. (Maybe) if a conversation is especially important, consider asking if you can record it. I found that at EAG people were surprisingly open to this (as long as I promised not to share), but I don't know if it's because they would have been uncomfortable saying no.

As a minor point, I want to push back a tiny bit on asking native speakers to do things for us.
I'm a bit afraid it would add a trivial inconvenience for native speakers to talk to non-native speakers, and increase instead of reduce the EA English-native "bias". (Which I agree can be a thing, as another non-native speaker told me once: "native speakers just take up more space").
As an anecdote: one of the most valuable conversations I had (if not the most valuable) was with a very popular/senior EA at EAG, who had just had an absurd amount of 1-1s that day and was visibly pretty tired. I'm afraid that if they also had to think "Lorenzo is not a native speaker, I'll need to pay attention to that", they would maybe have scheduled their last 1-1 of the day with a native speaker instead :/. I'm afraid that in day-to-day interactions it could make an even larger difference.

In general, I, very personally, would not want most people to apply the post recommendations when talking to me, unless after 5-10 minutes the conversation feels limited by the language barrier.
Some tips I would personally prefer are:
 - the one from Kirsten: "repeat the exact same words in the same order". I sometimes get confused when I ask someone to repeat something, and they say something different. I spend the next 30 seconds trying both to deduce what was the original thing they said in the first place and to follow the conversation at the same time.
- "fake an American accent" heard it from someone at EAGx, if you have a very British accent it might be worth trying for fun.
- If you can't understand something I said, please tell me, and don't feel bad. I know my accent is pretty bad, it's my fault, not yours. 

On:

Be aware and try to control the voice inside of your head (which I am highly confident is there even if you don't want to admit it^[2]) that says "ugh, this person who clearly isn't understanding me seems slow and stupid. I don't want to waste my time with them"

Embarrassingly, I'm also guilty of something related to this.
On one occasion I think I greatly overestimated how strongly English fluency would correlate with achieving results, and overestimated the counterfactual impact of helping someone with a project because I thought their English was worse than mine.
On a separate occasion, I greatly underestimated the impact of a different opportunity because I thought being a non-native speaker would limit any contribution I could make.
So in general I now consciously try to update less on English fluency when making predictions.

Luca -- thank you for this. I totally agree. We native English-speakers are often quite unaware, and unwittingly exclusionary and alienating, when we fall into these ways of speaking.

This can happen even across different English-speaking countries! I grew up in the US, but have also lived in the UK and Australia, and there can be significant failures of communication even across these academic cultures, based on different dialects being spoken too quickly, with too many culture-specific references.

I notice that many EAs adopt a sort of in-group 'EA dialect' that involves very fast speaking rate, staccato pace (fast-pause-fast-pause), and slightly over-annunciated consonants, as some sort of IQ-signal. 

It sounds like the way that the hyper-intelligent Mentats from 'Dune' might speak, if they've had a bit too much Juice of Sapho.

That EA dialect is pretty hard to understand for non-native speakers, I imagine. I teach a class on Psychology of EA, which includes many undergrads who don't have English as their first language, and they even find it hard to follow  YouTube videos of many talks at EA events.

If we want EA to be a global movement that's accessible (at least) to the 1.5-ish billion people who speak English as a second language, and not just to the 400 million-ish people who speak it as their first language, we need to be more aware of the issues you raise.

I think these tips are generally good. But one more we might experiment with, especially when online/remote:

Use voice-to-text speech recognition, and possibly also even translation.

My experience/impression is that this takes a bit of setup (good microphones, clear speaking voice, quiet setting etc), but actually works surprisingly well. I suspect we might be coming to the point soon where this actually leads to better and faster conversation between people with different native languages.

It’s hard to process new and complex ideas while also dealing with a linguistic burden (having to translate a foreign language, or speak your own language extra precisely, and adapt to unfamiliar pronunciations.)

Using this machine tech may be somewhat awkward and seem less natural. But the benefit lower cognitive load, and being able to focus onto issues may outweigh this.