So...this response is really guesswork...
I have not been in a similar situation but build scorm
stuff and have for a long time in several types of software, so I have some ideas as to what is happening. I think
it depends a lot on what type of authoring software is being used and how it is set up.
If they are doing what I suspect they are, when a score is submitted, an email gets sent with data, most likely not anonymized. Then the data is collated for evaluation.
Lectora creators have more granular control over, well, everything. I don't have a copy these days, so can't confirm but I think this control may extend to how/what results get sent. I doubt they are using Camtasia to do this. Captivate allows less granular control and is just send it or not, there may be ways to choose to anonymize but I don't see them.
If the module is built in captivate and you run it in your moodle, you could pull the scores out (sql
report) yourself and evaluate because it actually breaks results down by question (which is most likely the base of any game). If it is built in Lectora, this is a bit trickier because the reporting comes through in a data string separated by semi-colons (there may be away to change this but I don't know it). I can't speak to any others (other than Camtasia reporting is pretty ugly). There are probably ways to select variables and put them in an email with no student data for most softwares at all but they are probably more work. I could probably do it in Captivate but thinking about it makes my eye twitch and gives me a bit of a headache (so I'm not going to).
A shorter answer would be: Your best bet is probably to ask them what they do, what the data they get looks like. You could ask that the data be sent to you instead of them. You could ask that they do something to ensure data privacy.