Good to see you posting again. You're definitely better off with sidewalk strips than crack houses!
From a legal standpoint, it depends on the town.
In many towns the city owns the street and the property up to and including the sidewalk. But in other areas where the town only has an easement or right of way for the sidewalk strips then they are technically the home owner's property.
Asking every person for permission to detect a city owned sidewalk strip is not a very time effective way to do it. And if you go up to doors soliciting permission, sooner or later you're going to get someone calling the cops on you because you're a stranger they don't know at their door and they're afraid. But if you see somebody watching you from their property, or coming or going while you're detecting, it makes good sense to engage them and make sure they don't have a problem with you doing it in front of their house. If you are polite and non-threatening you might even get the OK for the yard too. If they do have a problem with it, be polite and get out of Dodge. I'd suggest leaving the block and getting out of view of that person.
Detecting these strips during low traffic times makes a lot of sense to me, the first couple hours after sunrise on the weekends are ideal.
Regardless of who legally owns it, most homeowners think of that strip as their property. They mow it, they maintain it.
If someone calls a cop because they see you digging in front of their house, how do you think the officer is going to react? You can bet he (or she) is going to see you as being the problem, regardless of legality. That cop doesn't want to be bothered with stuff like this and he's going to think that if he lets you keep doing this other people are going to be complaining too.
Let's take it a step further. Suppose the complaining property owner contacts a city councilman, or the mayor. Or maybe he is the mayor. Next thing you know you've got a movement going to ban these guys who are "digging holes out in front of our yards". Sound far fetched? Maybe. But that's how ordinances are created, from complaints made by citizens.
I'm not arguing against searching sidewalk strips, but bring along a strong helping of common sense if you do.