So, you used Flickr, and trusted Ya!hoo!! but you don't like Google. And you want stuff free. Fair enough, your choice, if slightly inconsistent.
Who said I trusted them? I used them because Google was barely a twinkle in someone's eye when I set the Yahoo acct up and if they've been able to extract any value from the crap and generally misleading data I set up my Yahoo account with, getting on for 15 to 20 years ago, then jolly good luck to them. Oh yeah the Flickr acct, which was only 10 years old, was likewise an utter fabrication as far as personal details went.
Now we've finished with the obligatory name-calling I'll repeat my request for a non-Google alternative to Flickr. And no it doesn't have to be free, but very, very cheap would be good.
I've re-read what I said, and can't find any name calling there, sorry.
For some reason, you are happy creating a fictitious Yahoo account but not a fictitious Google account. Let's be honest here, both companies will try to "monetise" what you use them for, if mail, then they have access to substantial hard data, if just pictures, much less so. All I'm highlighting is that you are being somewhat inconsistent, which is your prerogative. I can fully understand why anyone feels uncomfortable about how the megacorps handle and use the data we give them, while I'm informed and comfortable with the deal I don't expect everyone to be. While it is only my opinion that Facebook and Ya!hoo! are the more evil players, it is a supportable argument.
If you want private, then you can run your own hosting server, there are some cheapish deals around, but expect it to cost you at least £50 year, likely more. Of course you won't have the security that the GoogleHoo provide, but as we all know that comes with strings. Dropbox is probably the sort of storage service you are after, or maybe smugmug for photos.