Suppose someone uses one of your photos someplace. You wail to your friends about the unfairness of it all, and someone suggests that you issue a DMCA takedown against the evil someone. So, you do that, jumping on the relevant web site, finding their DMCA takedown page, and clicking rapidly past all the legal boilerplate. You mash SUBMIT and grin wildly. Victory, surely, is yours! Your photograph is taken down.
A little later, the evil someone files some mysterious and no doubt illegal "counter notice" and the photograph re-appears! The web site becomes thoroughly unhelpful, having clearly been subverted by the evil someone, and now apparently you have to either go to court, or give up. This is the terrible! You have a sad! Your friends are baffled, especially the idiot who told you to file a DMCA notice! It is a conspiracy against you, and all photographers.
Except, actually, no it is not.
The DMCA takedown is step one of a legal process, about which you would know quite a bit if you hadn't clicked past the legal boilerplate in a hurry to get to the SUBMIT button. The thumbnail sketch of how it actually works is like this:
You file a notice with a web site. The web site arranges for the content to be taken down, and notifies the alleged infringer.
If the alleged infringer folds, you're done. This is the usual outcome.
If not, the alleged infringer files a counter notice, possibly because they are a large corporation with deep pockets, or possibly because they know more about copyright law than you do, and they're actually in the clear in their use of your photograph. At this point the web site you filed the takedown with will generally reinstate the content. This is when your photo reappears.
Fun fact: I had a guy tell me "it can't be fair use, because you stole it!" which is some kind of galactic-level nonsensical sentence.
It is not twitter's job, nor google's, nor Facebook's, to work out who really owns the photograph. These entities are just serving notices and acting in accordance with the law. The actual resolution of whether someone has or has not infringed your copyright is a matter for the court system, if you and the alleged infringer don't work it out yourselves.
Once the counter notice is filed, you have two options: you can fold, or you can go to court. If do you decide to go to court, there may be a fairly short clock running, you may have to decide to retain a lawyer within a couple of days. Don't try this without a lawyer, by the way. Really. Maybe try representing yourself on a murder charge, but don't give it a shot in a copyright case.
Filing a counter notice does commit the filer (the alleged infringer) to the next step, which is court. If the original filer of the DMCA takedown does not fold, costs begin right about here. So why would anyone file a counter notice? Who really knows, but among the possible motivations: there is no infringement involved (e.g. fair use), or the infringer is prepared to take a risk of losing probably after estimating that you're bluffing.
Almost everyone who files a DMCA notice is bluffing, whether they know it or not. The process leads, pretty directly, to court. If you're not prepared to go to court to not only prove infringement but also to prove your ownership of the copyright, you are bluffing. Both of these things are probably a lot more complicated than you imagine, and there are a lot of pitfalls. I am not a lawyer, and cannot advise beyond "ouch, complicated and risky."
Is it safe to bluff? Well, yeah, in practice pretty much. You can just fold if a counter notice comes back. There may be some ways you can get hammered here, but the normally if you filed a DMCA takedown, you will just fold at this point, and the alleged infringer wins.
Technically, though, you probably shouldn't be submitting the notice in the first place if you're not prepared to back it up in court. The DMCA is not a toy, it is not a tool for harassing people you're angry with, and it is not a hammer of automatic victory for copyright holders.
It's just step one in a legal process that leads in a short number of well-defined steps to the courtroom.