Do you think it will be a problem if I move sometimes to the other location? I mean the private server will stay in one place but I guess I can just access it online.
What tool to use to have bookmarks on our own server? I mean how to do that? Generally speaking , I mean.
You need a self hosted server using a SSL certificate of course of which there are many free ones. Wix dot com comes to mind.
Save your bookmarks as a html file in: Bookmarks/Manage Bookmark/Import and Export/Export bookmarks to html
Move it to the server
Keep it up to date.
I use this method not for bookmarks but for certain pages I want to keep private. The first step in this page is checking if the IP address is mine.
Could be a pain if your IP changes frequently.
EDITED later:
I just thought of problem. How to keep it up to date. I presume at this point all your bookmarks in the browser have been deleted. Now you need to add a new one. Better know your html.
Maybe I made a mistake about Wix above. Seems they are not free.