This is my personal answer:
As a NEW fellow getting out, I would advocate you eventually owning your own practice or getting partnership into a practice (latter is harder to do as it requires trust and assuming they'll give you true partnership). With that being said, it is obvious that you cannot open something immediately after training given level of debt and inexperience.
So, you will need most likely employment for 2-3 years.
Now here's the difference between PP and Hospital.
Hospital Pros:
Pays more up front than PP. Covers malpractice and tail. Good benefits, retirement, steady job, etc. Great for experience, referrals, etc.
Hospital Cons:
1 - Administration. I cannot stress enough about this. This is the most annoying part. You are not a respected Physician, you're merely a Ford assembly line worker in their eyes. They will tell you what to do, and if you don't want to do it, now comes the passive aggressiveness and meetings. If they have a certain vision, that's it. They'll falsely have a meeting with you telling you what your thoughts are, you're really supposed to say it's a "great idea" and you're "willing with work it."
2- Everything is about productivity; ie hamster going around the wheel. Look around, see the physicians who are 45, chances are, they're exhausted, miserable, and feel "Stuck." You also aren't getting any benefit to this work; you're not owning anything or building equity in anything. Clock in/clock out mentality.
3 - Ultimately a scam. First 1-2 years are "golden cuffs." Solid base to start you off, with productivity goals you're never really made to meet. Once you're off the guarantee, you find yourself working like a dog, constantly to meet/chase "RVUs." That golden 4-6 week vacation? Lasted 2 years, because after that, taking vacation cuts into your RVU productivity numbers.
4 - Some hospitals want call/inpatient consults. Those are brutal, you're better off bailing from that gig.
5 - Sometimes you're forced to work with people you don't agree with clinically.
6 - Inability to control variables or anything with the practice. What you want doesn't mean you'll get it, or it could take forever. Best part? Ultimately when things flop, you will be blamed for why "things aren't growing."
7 - Doesn't matter how much money you bring in, you'll constantly be told you're not bringing in enough revenue. They have to keep that pressure on you to keep producing and working to bring them in money.
Private Practice Pros:
1 - More voice.
2 - What you want, so long as its reasonable in the budget, will happen quickly.
3 - More innovative procedures I've noticed.
4 - No Administration.
5 - You are the Physician amongst the others who own the practice or are employed by the practice. Your voice is often more important. (Assuming this a non-private equity backed gig).
Private Practice Cons:
1 - Less money than Hospital.
2 - I notice many won't cover tail malpractice.
3 - Promise of partnership isn't necessarily true and you can be cut right before you're scheduled to become a partner or have the ability to buy in.
4 - No access to books obviously unless a partner.
At the end of the day; get your experience, then open up your own gig.