Daniell, the situation you describe would only come about if the infringer issues a counter-notice. Essentially, this says that the original notice (the one that you issue) is wrong. In those cases, you are right that there would need to be some way of resolving the matter.
But those cases are very rare. If you are satisfied that you own the copyright, then it is highly unlikely that the infringer would claim otherwise. In the vast majority of cases, one of two things happen:
The hosting company, etc. accepts your claim (because there is no counter notice) and removes the content.
The hosting company ignores your claim (because they are inefficient, lazy, indifferent, etc) and nothing happens.
In the second case, you then have the possibility to pursue it further (to the extent of taking legal action in the courts if you think it worth the effort).
In my experience, about half the time you will succeed straight away in getting the content removed. In the other cases, you won't get it removed, but there is a good chance of getting the search engines to remove the offending pages from their indexes, which, if nothing else, discourages the perpetrator from doing it again.