When does someone entering another persons property become a trespasser?

I depends on your local and state laws and statutes. From my experience, a person is not trespassing until they have been informed by signage or in person they are not allowed to be in the property or area. If the person refuses to leave or returns after they have been informed they are trespassing.

Added: Also, if they enter upon the property with the INTENT to commit a criminal act, they become a 'trespasser' the moment they set foot on the property.