Files aren't magically stored forever.

They could have the sampling machine write to an SD card and mail that back, or store it with a one-time-use link like pwpush.com, or have it expire & self-delete after N days like WeTransfer.

You of course have to trust that they aren't secretly keeping their own copy on S3 forever, but that'll always be a risk, and it can probably be contractually enforced and audited.

Much like many cameras process and discard the RAW sensor data after processing to pixel data if not further compressing and discarding the processed pixel data to a lossy image format, even more so the raw BAM sequencing read data is vastly larger than processed VCF files. Even many companies that retain that data are liable to archive it offline rather than keep the raw reads permanently accessible online. There are real costs involved and a business case is needed for keeping that data. Especially with a decent privacy policy or regulations storing 100G or multiples of that online for free and downloading on demand is a significant compared to the entire cost of sequencing.

"Not keeping it on their side" ... well. If they sequenced it, they have data in their computer, right? How could you avoid that? I don't see this as possible, it depends on trust"

First it depends on the contract, if it states they have the right, then they can and will legally use it.

If the contract would say no - then they would have a much harder time making use of that data, as it would be illegal.