I run Ollama with Open WebUI at home.
A) the containers they run in by default can’t access the Internet, but they are provided access if we turn on web search or want to download new models. Ollama and Open WebUI are fairly popular products and I haven’t seen any evidence of nefarious activity so far.
B) they create a profile on me and my family members that use them, by design. We can add sensitive documents that the models can use.
C) they are restricted by what we type and the documents we provide.
Ollama and Open WebUI, as far as I know, are just open source software projects created to run pre-trained models, and have the same business model as many other open source projects on Github.
The models themselves come from Google, Meta and others. Have a look at all the models available on Hugging Face. The models themselves are just binary files. They’ve been trained and there are no ongoing costs to use them apart from energy your computer uses to run them.