This post from DuckDuckGo [https://spreadprivacy.com/google-filter-bubble-study/] explains their methodology for investigating self-serving bias in Google searches. Simply put, the approach asked different individuals from different locations to execute the same searches. This was done with and without enabling privacy mode. The dependent variable was the top searches returned and these hits were analyzed using a procedure that measured the uniqueness of the hits across participants. I have tried to approximate what I thought such a methodology would be using open and private mode without much finding much difference. The DuckDuckGo results explain why this might be as even private mode showed bias. Another difference was probably the search term used. I used apple (would one see computers or fruit). They used vaccinations, gun control, and immigration.
It would be interesting to duplicate this methodology and add different search engines as an additional variable. This would be a nice project for a college course.