yes the salmonella that reptiles tend to carry is only in the poop, not the mouth, and its the non-enteric strains, so there wont be any horrific infections at all, just a bit of an upset tummy if you ingested some poop lol!
As for bites, their mouths will contain some bacteria, like ours but nothing too horrible. You should wash any bite, as said before some alcohol wipes, and let it heal, if it becomes really inflamed or begins to pus go to the Dr for some antibiotic ointment.
i think going to A & E about it is overboard really, i get scratched and bitten all the time by my cats (whos claws are MUCH dirtier than any part of a dragon) and i have never been told to go to A&E lol.