According to the DMCA, in order to avoid potential liability, the provider is supposed to take down the content "expeditiously" upon receiving a notice and not restore it for 10 days, even if you file a counter-notification immediately. In theory, someone knowingly filing a false claim is then liable for damages, but good luck collecting on that.
In practice, though, the one time I sent a DMCA notice to a hosting company, they tried to mediate between me and the site owner rather than immediately taking anything down.
Note that as far as I know, non-US companies will usually comply with DMCA requests.
"Note that as far as I know, non-US companies will usually comply with DMCA requests."
No they won't if operating outside US jurisdiction. As a UK hoster we won't and don't recognise DMCA requests. We'll only remove content upon a receipt of a properly filed request/complaint by English lawyers (I work for a company registered in England).
In practice, though, the one time I sent a DMCA notice to a hosting company, they tried to mediate between me and the site owner rather than immediately taking anything down.
Note that as far as I know, non-US companies will usually comply with DMCA requests.