Regarding your title, the short answer is "no." The long answer's a bit involved, so I'm going to cover your other questions before the general systems description.
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Does this mean they didn't actually report me?
While it's very possible they didn't, it doesn't necessarily mean that. Simply put, there are no systems in place that allow for immediate feedback on reports against you.
Or do you only get notified when you have been reported a lot?
Specifically, it's when you've been reported a relatively large amount in a relatively short time.
Okay, so here's a quick-ish rundown on the reporting system, and the related in-client pop-up warnings.
Firstly, the system doesn't measure the flat number of reports, but rather operates based on the number of reported games compared to your total number of games. A case is built when you've been reported in 12-25% of your games. The up-shot of this is that a single report in a game is worth the same as the infamous x9 report.
The pop-up warnings operate concurrent with this system. Specifically, they trigger when there is a "spike" in reports against you. If, for example, you are reported in about 3% of your games on average, but in the past week or so you've been reported in about 7% of your games, a warning may be triggered for you.
Secondly, those warnings should never show up in the lobby of or immediately after the game that triggers them. There is supposed to be a delay between the triggering report and the display of the warning. It's unknown (by players, anyway) how long that delay is, or how it's measured, but I personally suspect it's somewhat variable.
The goal of this is explicitly to ensure that players don't know who reported them, which itself is intended to prevent reprisals of varying sorts.
So what does that mean for those times where people comment on being reported when you reported them?
It means they've probably been acting toxic for - at least - that whole day. Or that they're decent guessers.