There are several "good" LLMs trained on open datasets like FineWeb, LAION, DataComp, etc. They are still "ethically dubious", but at least they can be downloaded, analyzed, filtered, and so on. Unfortunately businesses are keeping datasets and training code as a competitive advantage, even "Open"AI stopped publishing them when they saw an opportunity to make money.

What is the concern with only having weights? It's not abritrary code exectution

Unless one plugs it into an agent... which is kind of the use we expect right now.

Accessing the web, or even web searches, is already equivalent to arbitrary code execution: an LLM could decide to, for example, summarize and compress some context full of trade secrets, then proceed to "search" for it, sending it to wherever it has access to.

Agents can also be allowed to run local commands... again a use we kind of want now ("hey Google, open my alarms" on a smartphone).

There are several parts to the "spying" risk:

Sending private data to a third party server for the model to process it... well, you just sent it, game over. Use local models, or machines (hopefully) under your control, or ones you trust (AWS? Azure? GCP?... maybe).

All LLM models are a black box, the only way to make an educated guess about their risk, is to compare the training data and procedure, to the evaluation data of the final model. There is still a risk of hallucinations and deceival, but it can be quantified to some degree.

DeepSeek uses a "Mixture of Experts" approach to reduce computational load... which is great, as long as you trust the "Experts" they use. Since the LLM that was released for free, is still a black box, and there is no way to verify which "Experts" were used to train it, there is also no way to know whether some of those "Experts" might or might not be trained to behave in a malicious way under some specific conditions. It could as easily be a Troyan Horse with little chance of getting detected until it's too late.

it's being trained on the output of other LLMs, which makes it much more cheap but, to me it seems, also even less trustworthy

The feedback degradation of an LLM happens when it gets fed its own output as part of the training data. We don't exactly know what training data was used for DeepSeek, but as long as it was generated by some different LLM, there would be little risk of a feedback reinforcement loop.

Generally speaking, I would run the DeepSeek LLM in an isolated environment, but not trust it to be integrated in any sort of non-sandboxed agent. The downloadable smartphone app, is possibly "safe" as long as you restrict the hell out of it, don't let it access anything on its own, and don't feed it anything remotely sensitive.