I ride English so I tighten the girth from both sides anyway. Having double elastic ends on the girth is a big help. I also drape the girth over the saddle without it being attached and the stirrups are run up, so it isn't an issue on which side I put the saddle.
I could mount Conny from either side and often did, but haven't tried it with JJ or Casper. I don't really think they'd care, though. It's a bit awkward for me and I'm a lot clumsier mounting from the right, but it's not too major of an adjustment for the horse. A second or two of, 'Whafu are you doing?' and then they settle.