Where I have been in the park, I don't know how you could.
I found it very difficult to do with my long time fishing buddy, what with lines and rods flying around. We took to "Leap froging" each other. One of us would drop in, the other would wander upstream 50 yards or so and drop in. Fish up to near the upstream guy and leap again.
Round home here, I fish off the bank a lot for Catfish, Sauger and such. I like to have, and like to give space, but several of the spots are just so good they are, by long standing tradition, "shared spots".
Meaning if someone can fit in, they do.
I have met a lot of good folks this way and learned much from them. Often fishing with them all day and never exchanging names, yet running across them several times a season and doing it again. Share the beer, share the bait/lures, sometimes share the fish. And of course, share the Lies, uh, I mean Stories.
I have yet to have any real problems, though I have been uncomfortable a time or two when out numbered by drunk teenagers while fishing by myself after dark.