Public or Private Equity

Sounds like you don't know the industry too well if you have to ask that question. I would recommend that you read up about what each is and how people invest in each. Then once you get a better grasp on what each does you should be able to answer your own question. Also, if you have to ask, then it is more than likely that getting an IB/PE M&A internship would be a very difficult time for you. 

But, since you asked, private equity. Public equity can be anyone with a Robinhood account that wants to buy stock, but private equity invests in companies through either minority investments or through M&A. If you're able to land a role in private equity as a sophomore, then I would strongly advise you to stay put in that internship for as long as possible and try to convert it to either a return offer for the following summer (and subsequent FT role) or to another PE internship. Most people only go into IB because of the exit to places like PE so if you can go straight there, even better.