A trespasser is trespassed from a property by law enforcement at the request of the property owner. This is called a criminal trespass.
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In most places, a property owner must ask law enforcement to trespass the person off of the property before someone is considered legally or criminally trespassing. In most places a warning, either verbal or by sign or other means, must be given before a person can be criminally trespassed, but that is not automatic as the property owner may choose to not enforce it.

not true. I work in contract security training guards. have for years.

Trespassing generally only becomes illegal/criminal when the trespasser becomes aware that they are in fact trespassing. For examples, somebody wandering onto private property from public property, they could become aware of the fact. For example, if you have to jump a fence to get to where you were, or you passed a 'no trespassing' sign, or if someone is telling you you're trespassing; or, for example, you're there to vandalize stuff, or maybe shoplift.

The act of "trespassing" somebody is simply informing someone that they're presently trespassing. You don't have to be a cop to trespass someone; property owners have the right (and, generally, the obligation,) to control whose accessing their property and for what purposes. you can be asked to leave by a property owner or the agents thereof at any time, and that act of being asked is called "trespassing".

As for when it can be enforced... that's when the person is aware of their presence being unwelcome. Doesn't matter if the property owner is there or not, exactly. no warnings have to be given, for example, if it's reasonable that someone shouldn't be there. for example, you intrude into a nuclear facility, we're cuffing you up and handing you off, no warnings given. Similarly, if a group is throwing a kegger in a private parking lot, it's generally unsafe to go out and warn the group.