Should a zika-infected person stay away from other people?

Probably no need: There is no evidence of person to person transmission of Zika, except maybe rarely by sexual contact. There is no known transmission by contact with a contaminated environment, shared utensils or food, toilets, etc. However, it's a new problem and new or different information could become available. But for the time being, the virus is not being transmitted at all in the US.

No: No. Zika is spread by mosquitoes so it is safe to be around other people if you are infected. However, it is important to prevent mosquito bites if you have been diagnosed with Zika, so use insect repellent, long sleeves, and avoid places with mosquitoes until you are no longer sick (about 1 week). Zika may also be transmitted by blood and body fluids. Cover any wounds and avoid unprotected sex.

Sort of "yes": May be "contagious". A mosquito can bite a recently infected person (during the first week or two of his infection) and then bite a non-infected person and thus give the virus to the second person. A contagious person can transmit the virus through some sort of sexual activities (details pending, as of Feb. 2016).