I am a complete novice with GIS, so I failed to notice the obvious.
The Coordinate Reference System (CRS) gave distances in degrees (I had EPSG 4269, which was the default). I wanted to work with metres.
When I entered a minimum distance of 500.0 apart from one another, that was impossible, so QGIS assigned only one point per polygon. When I removed this restriction, it still only put one point per polygon, so I think this was not the only problem.
I had to change my CRS to one that works with metres. I chose UTM, and found the appropriate zone (since I'm in Canada, I used this website). Then:
1) Right click on the layer
2) Click 'Save As'
3) Next to the CRS field, click 'Select CRS' (which is a small icon)
4) In the 'Filter' field, type the CRS you want (in my case, when I Googled 'UTM zone 14', I got EPSG 32614, and that's what I entered in 'Filter').
5) Click the result to make sure it's highlighted, then click 'OK'.
Note this might give you a save error. To get around this:
Next to 'File name', click Browse.
Navigate to a directory in which you would like to save your file. Click it. This enters the entire path into the 'File name' section.
Rename 'Untitled' to something meaningful, and save.