No, being tested positive for SARS-CoV-2 doesn’t mean the person is contagious. We know this because we can’t always culture live SARS-CoV-2 (that can still replicate in lab-grown cells) from a nasal/throat sample taken from an infected person.

Being contagious requires the infected person to shed large amounts of live viruses from the upper respiratory tract. So, a SARS-CoV-2-infected person is most contagious early on, when SARS-CoV-2 just entered the nose or throat.