Yes, you get a little high, for a qualified definition of what being “high” is (ex-pot smoker here). The “relaxant” effect is, in my experience, not because you get all, well, relaxed, with looser muscle tone, or slightly thicker-witted (drunkish/“stoned”) care for what’s going on out there.
It helps you think about far more pleasant and engaging topics. It’s a Distractant.* Remember–or, better, trust everyone who has said so–after the Novocaine you’ve got nothing to feel except water splashing around your throat, your jaw muscles a little tired from being stretched, and the slight sensations dimly felt from distant continents in the rest of your mouth.
What I’ve always appreciated about the effects of nitrous, and welcomed the opportunity since I don’t get high anymore, is what it shares with the high of marijuana: you start thinking really hard about other stuff (what the hell else you going to do lying there) which may include a dispassionate observation of what those hairy or freckled arms are up to, or realizing that that hum which made you think about your air-conditioner is coming from the drill (“huh, how 'bout that”)–basically not relaxing you per se, but distracting you by stoking your mental or emotional self absorption (especially groovy when you have music cranked up on headphones, which you should ask for if they offer it).
*I just invented this word. I should trademark it sell it to big pharm.