I don't know about Stoegers, but I have a Savage side-by-side 12-gauge with two triggers and I've never been bit by the other trigger.
Side-by-sides tend to be a little more reliable because they have completely separate trigger mechanisms, if one breaks or fails you still have the other. Most have different chokes on each barrel so you also have the choice of which choke to fire first. And you can fire both barrels at once if you really want to.
Over-unders are a little easier to operate because you don't have to remember to switch triggers, but the mechanism is slightly more complicated and you can't choose which barrel to fire first.
And I'm not sure if this is generally true, but the over-unders I've operated had ejectors which completely eject the spent shells, but my side-by-side does not and I have to pull them out by hand, so the O/U was quicker to reload.