Well, I've been very successful with Berghias, and not successful with peppermints. I have an 80 gallon tank that had 30-40 medium-sized aips and about twice as many small ones. I bought two Berghia, and six months later I had no aips and lots of Berghia. I traded and sold more than 40 Berghia, while scouring every local store for aip-infested rock (and in the Los Angeles area, there are a lot of stores).
Eventually (a year ago, actually), my Berghia population crashed (have no idea why). Unfortunately, I still had some of the "feeder" aiptasia I had purchased left in the tank (I figure I bought well over 1,000 aiptasia over a six month period!). So now, a year later, I have more aiptasia than I had when I bought my first Berghia. I've bought several berghai over the past six months or so, and although they ate quite a few aips, they never reproduced (which is when they become really effective in destroying an entire aiptasia population).
During this time I also bought 6 peppermint shrimp (all at once). All I can conclude is that my tank is shrimp-unfriendly, as all the shrimp were gone within a month, and there was no dent in the aiptasias.
I just bought 8 more Berghia, and I hope these will be more successful in reproducing (I've only had them a few days and I can already see a reduction in aiptasia in the area where I released them).
Finally, I've lost several SPS frags over the past year to my aip infestation. So, my suggestion to anyone who has an aiptasia problem, address it quickly (with whatever works), use overwhelming force (to quickly decimate the aip population), and never, ever, buy other folks' aips when yours are gone to feed your aip-eating critter (whatever that may be), because you just never know when you aip-eater might suddenly depart and leave you with a viable aip population (and even one aip is a viable population!).
Kevin