This strip is not ordinarily a right-of-way or easement.
In a platted area, it's city property. Bear in mind that the homeowner may be totally unaware of that, and that he mows it and keeps the weeds off of it. Myself, even knowing this, I don't detect it without permission. Even though I can legally trespass in my profession when working, I ask when I practically can. It just makes sense - and friends.
In less common areas (mostly outskirts or rural areas), it can be owned by the property owner.
Then they own it to the middle of the road (or less often to another line, like a section line or prior ownership). You can't detect there then. Just because the public has a right to travel it, does not give you a right to detect it, or pick asparagus for that matter.
Hope this clears this up some.
Disclaimer: This is not legal or boundary advice; always seek site specific advice from qualified professionals.