As a replacement for Chrome on your PC, Brave takes the second approach, and performs the evaluation locally (pulling down a fresh list every so often for up-to-date comparisons). You'll have to check with Vivaldi on which approach they take.
Brave has all of the phone-home-to-Google logic removed. If you ever spot anything at all that gives you concern, do feel free to let us know.
Sampson (Brave Developer Relations)
After having just posted here, it was...disconcerting. I'd like to give you guys the benefit of the doubt, but you may want to check a fresh update & relaunch from 0.23 to 0.24. I doubt I could reproduce (I've already clicked past it, and it was only on the first re-launch), but for a moment I was pretty confused about what browser was open (in fact, the only reason I'm certain it's Brave is because I just double-checked my Chrome and it's still on 0.68). Brave 0.24, V8 6.9.427.23, rev f657f15, macOS x64 (10.11.6 El Capitan), OS Release 15.6.0, Brave Sync v1.4.2, libchromiumcontent 69.0.3497.100.
I'm posting this comment on Brave, having switched over last week for most of my personal browsing. Generally like it a lot; there are a couple warts (it won't open S3 URLs from the AWS web interface, for example), but it does most of what I need in a browser.