I certainly would not post it for anyone to examine. I did apply for my list of decorations, and got a nice list, along with a few ribbons and one medal that I left active service too soon to get in person. That should shut down any nosey types. Worked recently for an @sshole swabbie nuc-sub type that was certainly the kind to question my service. Too bad for him, I just ignored his BS sub stories and made it clear I was not impressed. He never took the bait and asked if I had served..........
Replacement copies are awful hard to get from your service branch. If they’re recorded at the courthouse you can get a legally certified copy in very short order for about $12 (your county may vary).
It is too valuable to lose. Consider recording it, unless you have something you’d rather not disclose.
I would think the only reason to have it filed, would be if you routinely referred others to it as a job requirement, such as for a defense contractor position. Even still, there are probably other ways to share that information.
One last point, my DD214 had my SSN on it, as that was routinely used in most forms of documentation and IDs, most likely prior to the year 2000. I wouldn't want to have that available as public knowledge.