How do you as a product marketer create brand messaging that appeals to three highly distinct segments?

Messaging that speaks to everyone rarely speaks to anyone well. I’d prioritize your top segment to focus on, make sure your brand messaging resonates with them, and ensure your other customer segments can find messaging for them via pathing on your site or other targeted campaigns.

I've seen most marketplaces have their first message be targeted toward the demand side and then have the supply sides be secondary way at the bottom. 

Examples:
https://www.doordash.com/
https://www.instacart.com/

https://www.imperfectproduce.com/

If you're really supply constrained, you might change your focus there. You can see that Uber and Lyft's sites focus on drivers right now:

https://www.uber.com/

https://www.lyft.com/

I assume by segments you mean industry/vertical and/or company size (SMB vs large enterprise, for example) - in that case I’d consider running methodical research work (see the other question answer on 4 recommended approaches) in different tracks aligned to each of the 3 segments, ensuring there’s a common umbrella message and thread through it all so it doesn’t sound like 3 different companies. Anchor in each segment and bring the message about your brand to each segment rather than the other way around - you’re too susceptible to confirmation bias that way IMO. Also, brand messaging is different from product messaging - the brand is more about who you are and what feeling you what to evoke/what associations people make with your brand from customers/prospects vs. what your product/offering does and how it helps customers. They have to be coherent and ladder up, but conflating the two only causes trouble.

When constructing your brand and messaging hierarchy it's valuable to have a 50 thousand foot view of the position for what your company represents, that helps anchor all of your audiences to what you do and what you represent. It also helps align the unique value prop for each audience to a central set of company values. For instance, does your brand represent innovation, vertical expertise, ease of use, etc. But when you dig in a layer deeper it's important to consider how that messaging will be used in your demand gen and sales plays.

It's extremely challenging to be all things to all people when it comes to specific objectives you're trying to hit. I would advise against trying to have one message for all of your products that is evergreen for each audience and all campaigns.

We generally take a product development type approach to building our messaging for our various products and audiences. That is, we build it iteratively around a specific audience need using a Feature Positioning framework that starts by us asking "Who is the audience" we're creating this for. It's okay to have more than one version of that Feature Positioning document for the product if it's consumed by two different audiences. It's much easier and more efficient to build that way when you're not trying to stress test how that message would resonate for other audiences too. Keep it simple.

In your instance (some form of marketplace that sells some sort of widget presumably) it may be a vertical based marketplace that is highly specialized, easiest to use, broadest selection, etc. However, the benefits of your marketplace will likely be wildly different depending on your audiences role in it (buyer, seller). As you go deeper into targeted campaigns that have specific goals (i.e., increase buyers in the marketplace, increase attach rate to certain categories, attract more sellers, etc) make sure your highly unique value props are aligned to those goals. 

Great question! We faced a similar challenge a few years ago for one of our suite of solutions. It may boil down to how critical each of those three segments are to your business. Based on your question - sounds like all three are important (otherwise I'd recommend optimizing for the most critical segment). Assuming you have to address all three segments, you have a few choices: Craft your brand messaging to include (1) the lowest common denominator for all three segments (2) the most critical components for each of the segments. Clearly pros and cons to both approaches in terms of breadth vs. depth of potential appeal. If you have the luxury of time (!), good to perhaps test out both options above with your audiences and see which one resonates better. In our case - I'll say we went with option #2 above initially but as our suite expanded, we focused our messaging and products much more to a specific audience  

It is recommended to start with one segment to test/iterate, to completely understand the buyer needs/value proposition and then build messaging for the other two segments. If it’s not unique for each segment, then it will be hard to position one overarching message. It will be challenging at first, but once you have the value proposition for each segment, you will naturally see the synergies and patterns.

If the message is too broad, it wouldn’t mean anything to any specific persona. Always test the message with Competition. If you change your company to a competitor, can they say the same message? That’s the best way to validate your differentiated value.

Work backwards. Develop very targeted (10,000 foot) messaging for each of the three. Then analyze those stories and find the idea(s) you can use to tie them together, or reconcile them with one another. That is now your 30,000 foot story. As far as the market is concerned, that's what you started with, and from there the targeted messaging was derived, instead of the other way around. That 30,000 foot story can be where your brand message comes from, whithin which you have distinct value propositions for three segements.