Can there be any good bacteria on your hands?

No: We all have bacteria that live on our skin & usually do not cause any harm. These bacteria are in the mouth, nose, vagina, anal area . Washing hands prevents bacteria that don't actually belong on our body from getting in when we touch our eyes, nose or mouth or through cuts in our skin. Washing hands with soap &water won't remove any bacteria needed to stay healthy. Those are in our intestines.

Yes: The skin has a mixture of normal bacteria that colonize it, causing no disease. These include staph. Epidermidis, corynebacteria sp. And the yeast malassezia furfur. These modulate the microecology of the skin, releasing fatty acids to produce a hydrophobic milieu and lactic and propioic acids which maintain a low ph. Many bacteria secrete bacteriocins that inhibit other pathogenic bacteria.

No, there are not: The hands, and your skin in general, are colonized by many strains of bacteria, but they generally just inhabit the skin and do not cause infection. Your hands may come in contact with some bacteria which are bad (like staphylococcus), but there aren't any "good" bacteria in your skin, especially on your hands since everyone touches many objects during the day which may have bacteria on them.