I'm exclusively running Firefox, but I'd like to install the leading browser Chrome for troubleshooting purposes. Browsing-related data collection isn't a concern in this case, but I'm worried if the browser logs key input on the background or scans and uploads files or related data to their server without notifying the user.
Chrome has an "on-demand" virus capability provided by ESET. This also does a periodic scan of all your files. If it detects "malware" it will send meta-data of that file back to Google.
Chrome also can log keystrokes for the purpose of "spelling error correction" but I believe this is in opt-in feature.
For a full list of what data chrome collects (at least according to Google) check out their white paper.