In most countries people are only remained in custody because they are either a physical danger, or because they are constantly re-commiting the same crime and having them walking around just facilitates that.
People who are awaiting murder trials are routinely released from custody. (Although they may have to pay for it in some countries.)
In the U.S., it can really vary. It's pretty common for people to be held at bail even if they're not particularly dangerous or are only accused of petty crimes.
Also, in the U.S. system, I would say that it's generally much easier, from the perspective of the accuser, to bring charges against someone than it is to take them to civil court. Obviously, not all criminal complaints result in charges or a court case, but there's very little risk involved in accusing someone of a crime because unless you perjure yourself or clearly make a false or bad-faith accusation, there's no legal consequence for making a criminal accusation that doesn't pan out.
In the civil system, people are responsible for bringing lawsuits against someone. You have to file the paperwork, compile the evidence, and pay court fees. You have to pay for an attorney if you want to be represented by one. Though there are plenty of frivolous lawsuits, I think the amount of responsibility you take on when you file a suit helps discourage completely frivolous claims.
The only upside I could see to trying copyright infringement in criminal court would be that the accused would be entitled to legal representation. As it is, a lot of people never defend themselves against frivolous claims of infringement because hiring an attorney and fighting is more difficult than obeying a cease and desist.