Agamon
Adventurer
This is the type of situation where I check surprise in 1e, but I guess in 5e you're not meant to check surprise outside of special situations (scouting and ambushes).
Not that special in my game. The players I run are quite careful and try to do what they can to make sure they fight on their terms. And when they aren't, predatory monsters are often trying to do the same to them.
It would be unusual for both sides trying to sneak up on each other though, can't think of that coming up often. I'd just say they surprise each other without rolling and then normal combat begins.