The biggest is whether the pulling force is centered on the trailer. If not, the trailer will want to twist with respect to the direction or travel and more a bit crab wise. That will increase drag and ride wear.

Within reason, it doesn't matter as much if the trailer is off to the side, since it's drag is applied to a dropout which is close to the bike's central plane. The biggest effect of an occasional center trailer will be slightly different turning properties, though I don't think they'll be noticeable at normal speeds and maneuvers.

Anyway, I'm intrigued with the idea of a dog trailing you all day, yet never catching up.