No the model actually runs remotely. We would love to have some way to run a lightweight model on the user's machines but it's just not feasible given model size and amount of computation required.
This isn't disclosed on the linked page, nor is a privacy policy linked.
I think that parent’s tone is way too combative. But the “Pricing” page on your website[1] only says that the service is free for early users.
One Plan
It's pretty simple. Early users get free forever access to Codeium.
It does not say whether all future users will have to pay or only those who want additional features. The wording strongly suggests the former.