I'm a current fellow at a large program going into private practice in a couple of months. Here's my two cents.
1) I would recommend doing your own cases. Yes you learn how to cardiac cases in residency (I did ~100 pump cases in residency) but if you supervise and start a case with yourself, a resident, an attending and maybe even a tech around, you lose the learning that comes with starting cases on your own. My program has a mix between fast private practice style cases (2.5-3 hour door to door CABGs or AVRs) and slower-style academic surgeons. To be able to go into a room and start your a-line, go to sleep, toss in a central line, +/- swan, and insert and do an echo exam in 25-45 minutes while managing the hemodynamics on your own (granted with attending supervision) is a lot different than supervising. Don't get me wrong, I have friends doing fellowships where they supervise but depending on where you end up, you'll be much more prepared to knock out a pump case safely and efficiently under production pressure.
2) Get as much surgical variety as you can. Having just gone through the interview process (and listening to all my cofellows' stories) you want to be prepared to do everything. A private practice or academic group may desire to hire a fellowship trained person not just because they are increasing their volume, but because they're expanding their surgical services. This means go somewhere where you will do heart transplants, lung transplants, plenty of VADS, complex aortic surgery requiring CSF drains, and TAVR's. Having done robotic heart cases in residency and not in fellowship I feel like you can learn how to place coronary sinus catheters with a good foundation that fellowship provides. No matter what a places hiring needs are, you want to be confident you'll be able to handle the cases
3) Echo is of course a big part of the training. Supervising vs. doing your own cases may or may not effect what your numbers are. Don't forget though, all programs should be able to supply you with the 300 "interpreted"but they may differ on how many you "personally perform". For example, I went to a residency program where the fellows supervised which was great because on every one of their 6 OR months (stander for all fellowships) they could bounce between 1 and 2 rooms to look at the echoes. However on the 1 or 2 months of echo, they went to the cardiology reading room and learned echo post-exam. In doing my own cases I spend my 6 OR months doing 1 or 2 echoes for my own cases each day. On my 2 echo months, I go to every OR that has a probe in (~6/day) and do an exam so we usually get about 150-200 performed exams just from the echo months). Bottom line is that you're smart people that are professional test takers at this point. You'll be driven, learn the material and do well on the exam. Just ask the details on how the echo months work.
4) I'm not trying to solicit and have no affiliation with northeast programs but Tommy Burch, MD from BID has an incredible website called PTEMasters.com which has about 100 1-hour video lectures on echo which I feel are essential. It's normally $1000k a year but every July it goes on "special" for $300. I would HIGHLY recommend it.
5) No matter what you do be happy. Get well trained but don't kill yourself in the process. It's a busy year. By far the busiest you'll have in your training. Written boards in July, getting the wheels rolling on the job search and interviewing, oral boards and echo boards.
Good luck everyone!