Can someone contract an STD by sharing a drink with someone else, even if it is a while between drinks?

Are you talking about sharing saliva with someone who may have been positive and drank out of the same bottle?

You can only contract HIV through semen, breast milk, blood, pre-ejaculatory fluid and vaginal fluid. It would be highly unlikely to contract any other STD from drinking alcohol fromt he same bottle. Saliva is not a common mode of transmission.