My puppy used to be really nippy too and if it broke skin i'd usually just wash my hands right away and make sure to keep the punctures clean.
If you're worried about it you could put some polysporin on it until it heals as a preventative measure, but a doctors visit shouldn't be necessary unless you start to notice it looks infected.
If I went to the doctor every time my puppy broke skin I'd have been at the doctor several times a week, lol.
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