It's 100% absolutely not safe yet. You can effectively copy and paste Pliny prompts and pwn any of the frontier lab models. Anyone with a little time and creativity can tailor a unique one and set hidden text traps for AI browsers or agents, and depending on what access you've given the software it could be very dangerous.
When my local LLM (llama.cpp) needs to search, it opens DuckDuckGo in a new window, loads the result pages in tabs, extracts content with Readability.js, and feeds it back. You stay in the loop - can see what's loading, solve captchas if needed. Less autonomous than Comet/Playwright, with a narrower use-case, but also less risk.
Its still a prototype though: https://github.com/tbocek/llm-local-web-search