As I rule, I always offer to analyze after games, win or lose, just to see what someone who had gone through the game in real time with me had to say about it.

I just got back into tournaments after over ten years away, and the acceptance rate has gone down from almost 50% to around 10%.  Part of it is the scholastic set, who prefer to run the game through an engine and/or consult with their coach.

Do you analyze with your opponent in the engine era?  Is this passe now?  Would you accept an offer if your opponent wanted to?  Only after a win/loss?

Lastly, when I play standard chess here, the acceptance rate for analysis is 0%, even though chess.com offers a handy tool with which to do it.  Has anyone ever used it?

I'm only 13, playing in just 3-4 tournaments, when I was seven I was matched up against players rated 500-600 points higher than me, and 4 or 5 years older, so I lost 50% of the time, I was the only one on my team who analyzed. I can say analyzing doesn't hold thrill with kids. But, I do it and am one of the only I know who does. I've never had the opponent offer before, so...I'm not sure.

I love analyzing games that I just played! The one problem with it for me is that I often don't want to analyze the game with someone much younger than or someone much older than me. Therefore I get 100% analysis acceptance when offered. I think it's a great idea to see what could be improved in the game or what the other guy was thinking. You can also see during analysis what mistakes you where thinking. This helps avoid future mistakes. Well of course this is all just in my opinion.

If I finish my game relatively fast (otb) I will offer to analyze. One time, after losing to a 1800, he just started analyzing without asking me. However I would have wanted to analyze it anyway, and his advise was helpful

As a general rule, I accept any analysis offered, but I wouldn't analyse a game  that was lost by a single mistake (from either side).

As of chess.com, you should be aware that (unless I didn't find the good one) the multi-player analysis tool is only available to premium members (at least in live chess), so it could explain why none accepted your offers.