Last I checked, each cell service provider stored SMS content for some period of time. I don’t know anything about photos shared via SMS apps, though, even the technology used. Given how large the files are, hopefully service providers don’t store them, if only due to cost.
The MMS wikipedia page gives a good general rundown on how it works and has further links to what is replacing it—i.e. internet based instant messaging apps and RCS.
MMS was at the advent of colored cellphones and we have oh so nice OLEDs now. I think the world has moved on? Are there really still MMS protocols still in use?
I still have the odd exchange of SMS/MMS with a few who for one reason or another doesn’t use the same IM’s as me 
So don’t think it’s quite dead and buried yet.
I feel silly, because I read the pages for SMS and text messaging, but didn’t know the MMS initialism and/or skimmed over it. (It was late at night.) Is there any information on what service providers store?
The Technical description section explains it in broad terms and if you want more in depth information, you have the links and sources in the article. For a more specific answer beyond that you’ll probably have to contact your carrier and ask them.
But we should probably assume—with SMS/MMS being unencrypted and data storage becoming ever cheaper as time goes on—that they’re storing everything they legally can and have a use for.
So not to beat a dead horse here but it is always worth repeating: never send anything you want to keep private through unencrypted channels 
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For me it was a bit unnerving to discover the carriers themselves were outsourcing SMS routing. It makes sense, but was just not something I had considered before this article.