You are right to be worried. No service provided by Google is save against data mining. They are all used for data aggregation. In that respect indeed nothing is free. Your kids, and the school for that matter, are paying with their data.
We can only guess at the ways Google is using their data, but it includes:
- targeting advertisements and probably (and increasingly) content as well to your kids based on their
- creating filter bubbles so your kids only get to see what Google wants them to, they determine their view of the internet
- indirectly leaking personal data to (to us) unknown third parties (don’t know if they directly sell or provide data, like FB did)
- feed the data to their AI programs to have them learn from it and act upon it in current and future products
- inferring new information from the data they already have
About indirect data leaking: Even when a company doesn’t provide access to raw data, or sells it, information is leaked once you allow advertisers to target to specific users and groups. E.g. if an advertiser places an ad to ‘homosexual men above 40 in Alabama’, then it is easy for an advertiser to deduct that anyone loading that add and/or clicking through, is probably within that targeted group.
An argument you can make to your school management is that for them the use of Hangouts is also not free. Google will probably compile a detailed profile on the school’s performance, the competence of the teachers, the effectiveness of the teaching methods. And they can use that in the same ways as in above bullet points.
Not saying that they are actually doing this, but they could sublty move a school to buy certain products, have differentiated (pay the max) prices and provide the school’s performance data to government agencies and the like (maybe affecting subsidies the school is entitled to have).
In short: Both your kids and the school are the product once again, not the customer.