Hey bonjour,
For the pros vs cons of various search solutions available to Bubble users, see here.
Specifically to your question, preserving privacy rules with Bubble’s Native Algolia integration is not possible. However, it is possible to preserve privacy rules with Algolia if you build a plugin just the right way. I’ve done this and as far as I can tell, Omnisearch (my plugin) is the only privacy preserving search solution offering the scale and speed of Algolia or Typesense (or both if you desire).
See the Privacy Rules section of our demo - also check out our advanced filtering and faceting capabilities here.