And without interfering with or influencing the scene and therefore swaying the outcome? That's interesting since, as I mentioned, I'm only just dipping my toe in with street photography and I was under the impression (perhaps falsely) that this was somehow a guiding principle in the genre.
I think the general idea of "not interfering with the scene" in street photography is wrong from the start. When you're out shooting in the street, it mean you're having a physical presence yourself in those said scene. You may not see it in the picture, but you are definitely part of the streets at this moment in time. So when people say that you should capture scene that isn't influenced by your presence, that's not exactly the right way to say it.
Staged and "aware" moment are two different thing. Staging a scene is you, yourself, direct in a very particular way your subject, Aware moment are when people notice you and react to you.
Ideally speaking, you want to capture candid moment where people are unaware, where they didn't noticed you in Street photography, also referred as Candid. But realistically speaking, you having a physical presence at the moment you shoot, some people will react to your presence since you're acting differently from the crowd. I believe that's fine too, you can look at it as leaving your own mark in the scene without showing your likeliness in the stage. Being a die hard Candid is fine, but sometime you get that shot where people will notice you and you will still like it.
Sometime, you get their "revelation" face where they noticed your and snap out of it, I think it's hilarous. When you laugh after you generally get their angry face. That's fine in its own way ^^