I don't know...I dated someone in my med school and it wasn't a big deal at all. I mean if you're sleeping around in your class, then yeah, that's going to be gossip - don't be surprised. Frankly I don't think you should be doing anything that you'd be upset if it "got out". So choose your romances wisely. And what goes around comes around. Establish yourself as a person of character and integrity and people won't pin gossip on you as much. I guess my point is that for the most part, you can be above it all and it doesn't really affect you.
This concept applies to things other than romance as well. A list of advice for incoming M1's:
1) Don't tell people you want some competitive specialty. You'll save yourself criticism later if you change your mind or end up not being competitive for it.
2) Be friendly when people talk to you. It's fine to be solitary and hardcore or somewhat unapproachable (to avoid distraction or something) when you're in the library or in class or something, but when people do talk to you, be surprisingly friendly, pleasant, and nice. It goes a long way.
3) Share your study guides, notes, etc.! Honestly, it can do a lot to say you're a nice guy. Even once. If you make something awesome, be really cool, laid-back, and humble about giving it out to people and people will appreciate it and know you as a classmate rather than a competitor. People don't like gunners. Even gunners don't like gunners.
I will say that some amount of involvement in drama or gossip is pretty much unavoidable, as you said. People will know your business to some extent, so keep your ducks in a row to the extent that you care to.