The beach is unguarded and pretty much unmonitored. As long as you’re there during daylight hours, no one will bother you. I suggest swimming parallel to the shore juswt inside of the swim area marker bouys - you can practice sighting by swimming bouy to bouy. FYI one length of the beach is about .4 miles so 6 lengths (or 3 laps) roughly equals an ironman swim. Early AM hours are your best bet before boat traffic stirs up the water and things get a little choppy.
Edit: Oh, about your other question. The other places to legally OW swim are pretty much the other state parks. Deer Creek to the south, Buck Creek to the West, Indian Lake to the NW, Caeser Creek way south down by Cincy. I’ve known some people to swim at Hidden Lake in Grandview (5th avenue west of Riverside Drive) but that’s a private quarry and you’re theoretically at risk of getting scolded/thrown out (although no one I know ever has been - what are they going to do, come out after you?).
If you’re adventurous there are technically no signs prohibiting swimming at Antrim Park - although I’ve never known anyone to swim there at any time other than during the annual Columbus Int’l Tri. And if you’re REALLY adventurous, there’s a nice deep, clear, quarry in Hilliard at the NW intersection of Scioto Darby Road and I-270 that I’ve always been tempted to swim in - but it’s private too.
Let me know if you decide to take the plunge into either of the latter two. I’d love to but don’t have the guts to be the first.