When discussing locations in a city based on "side", I have never heard any other preposition besides "on". First thing that popped into my head (showing my age) is "On the south side of Chicago" from the song THE NIGHT CHICAGO DIED".

Other examples would be:
"On the other side of town..."
"Love on the wrong side of town.."
"On the sunny side of the street..."
"On golden pond..." (okay, that didn't actually apply, but Henry Fonda was the BOMB)

And why do we say "On the outskirts?"

Would you be "under the inskirts?"