What is the value of participating in proposal-writing efforts, for a postdoc researcher?
First, your titular question:


  What is the value of participating in proposal-writing efforts, for a postdoc researcher?


Especially in Europe, and when applying for faculty positions, the value is huge. Successfully playing the grant game is nowadays what keeps the lights on in many European universities, at least research-wise. Having a track record in this dimension can easily be viewed as even more desirable than a great research track record (although, of course, you will still need at least good publications).


  should I highlight this experience in my cv (i.e. provide more details about the bids in which I participated, my role, the outcome etc)


Yes, you should definitely highlight this somewhere in your application (not sure if the CV is the right place, though). Your letters may or may not be a good place, if LoR are considered important for the job you are applying to.


  If I have the chance to avoid being involved in proposal-writing efforts, should I?


This is an interesting question, and mainly depends on what your grant and research track records look like at the moment. If you have not actually been involved in grant proposals, I would certainly recommend getting this experience. If your research track record is lacking, I would focus on research. Realistically, you will need a good record in both dimensions to be competitive for faculty positions.


  Moreover, how can one's claims be verified? How do you prove that you are telling the truth and not over-selling yourself, regarding your participation and experience? Isn't this part of your cv a part that is less verifiable?


In the optimal case, you have won some grants in your own name. But even if this is not the case, the experienced recruiter knows what to look for and who to ask to figure out whether you are overselling yourself. Keep in mind that researchers are connected and EU projects are large - somebody in the hiring faculty will know somebody who was involved in your project and give them a ring to ask them whether you actually did what you claim you did. If that's also not possible, the recruiting faculty will just ask you a few questions regarding the claim they are interested in, and observe your reaction.