Usually they do act, I believe. If they are following the DMCA they take the alleged infringing content down and notify the party that uploaded that content.
If that party files a claim stating that it is not infringing the host is required to forward that claim you, and then wait a short time (something like 10 business days, but I don't remember the exact time).
At the end of that wait if you have not provided proof that you filed a lawsuit against the alleged infringer the host restores the content. If you do provide such proof the content stays down until the court resolves the matter.
If you do not provide proof that you have sued and the content goes back up and then later you do so, you would need to get a preliminary injunction or similar from the court ordering the host to take it down.
Some big platforms (Google definitely) use their own systems in parallel to DMCA, so your experience with them (on both the copyright owner side and the copyright infringer side) can be quite a bit different if you are trying to deal with an infringement through that, but if you go through the DMCA channel that will work.
If you aren't ready to sue though and the infringer counter claims the material will go back up. You can think of the purpose of the DMCA in the case user content hosting as being to get the host out of the loop.
The advantages for the copyright owner of going through DMCA first instead of just suing right off the bat are that (1) in the case of accidental infringement the infringer probably will not counter claim and so one simple DMCA claim by you gets the content taken down and resolves the matter, (2) if they do counter claim, you get a copy of that which includes contact information which lets you know who to sue, and (3) the content will stay down until the suit is resolved whereas if you sued first it would likely stay up until you could get the court to issue a preliminary injunction.