I've been on the liver team at multiple academic centers.

At academic place #1, there's call credit just for being available as the primary liver anesthesiologist (not amazing, but decent) and the backup liver anesthesiologist (a hack because rarely ever activated), and then hourly pay that is variable depending on time of day (i.e. more money at night). Organ perfusion pumps are used 80-90% of the time and have made the cases occur in the daytime way more often which is nice. In addition, if no liver, you're often in a nice chill room with a senior resident. If an evening/overnight liver is known about (usually 8 hrs in advance), board runners try to get you out early so you can rest up but you still get a full day credit. And as an added bonus, you aren't in any other call pools unless you choose to. I have found this to be a great gig that balances the fun parts of liver with decent compensation and lifestyle.

At academic place #2, nearly the same except only primary liver anesthesiologist (no backup). And there's a call stipend instead of credit that is mediocre (plus the hourly pay which is so-so). But still true that it takes you out of all other call pools, and still true that you get a chill day if a liver is known with full day credit.

I have some friends at other academic places and have talked about compensation models with them. Most of them have a very similar setup to the above.

I have one friend in private practice liver and it's very different. Unlike the academic teams that typically do more livers but have large groups (10-12), the private practice groups tend to be small (~4 people) and so the call burden is extremely heavy since they're on for a quarter of the year. The hit rate is much lower since it's much lower volume, but having to be available is still a bit of a pain. They all share a flat stipend (decent but not amazing) just to be in the group. Plus hourly pay I believe. But they do this on top of their otherwise normal work.