"...Is it possible to compete with each other..." Sure. Facebook or any of those sites will do nicely. Informal club practice matches scores aren't usually recorded by the club, but that doesn't mean you and your friends and family can't. No prizes or trophies, unless you organize it.
"...if you explain that you're competing in a 'postal match'..." That'll likely get other shooters interested in participating too. That's not a bad thing.
League matches are recorded and sometimes get posted on-line. Takes a volunteer to do it though.
Jump into a league(usually club vs club) if you can. It's challenging and great fun.