I have a Cooper / Eaton nightlight-receptacle TR7735W (manufactuer, amazon), which I replaced a standard receptacle with. The receptacle is not at the end of the run, so there's a line out to the next receptacle, and is not switched.
The old receptacle used the backstabs, but this new receptacle has the side screw plates. I wired the two hots to the two hot-side screws, the two neutrals to the one neutral-side screw (it has space for 2 wires), and the ground. I left the break-off tab intact. The switch and light work fine, and wiring is confirmed correct with my tester.
When I plugged in the vacuum, the light went off. I was able to get the light back on by pushing the vacuum plug to the side ever so slightly. Pushing it back the other way caused the light to turn off; I could repeat this to turn the light on or off repeatedly.
I admittedly haven't done any troubleshooting (didn't notice the problem with the vacuum plug until late), but is this indicative of a problem with the night light itself or just my wiring job? I'm thinking loose connection.