Technically Yes they could. However, a reputable barber would sterilize all their equipment between every customer.
Can A Person Get HIV If An Infected Person Was Getting A Haircut With Trimmers And He Was Nicked A Tiny Bit And The Next Person After Him Had His Beard Trimmed With The Same Trimmers And Also Was Nicked A Tiny Bit?
This is very dangerous, and it is quite possible a person could spread hiv and hepatitis this way
The AIDS virus has an EXTREMELY short life when exposed to oxygen. In open air, it can live all of about 5 minutes, give or take. While it's a possibility, it's extremely small. You would have more chance of getting head lice.