You know, I had not thought about that. I wonder how that came about in the old West? That is really interesting. They had lynch mobs, too, but it would seem that there may have not been much concern about those things then. I wonder when all that changed?
For one thing, that whole era of the "old West" you reference in your post dates back to a time when much of the West wasn't even formally part of the United States. Most of the West was comprised of terrotories that were U.S. possessions but had not yet been admitted into the Union as states. So for much of that era there was no legal system in place like we take for granted today. The U.S. military was probably the ultimate authority in these places, to the extent that it had any presence at all.
In addition, the term "outlaw" had a specific meaning for people that described not their criminal activities but their legal status under the applicable law at the time. We use the term "outlaw" nowadays to describe a criminal from that era who operated with no respect for the law, but the real application of the term was for a person who placed himself outside the protection of the law and therefore could be dealth with by authorities and/or law-abiding citizens without regard to the proprieties (e.g., a jury trial) of the U.S. legal system.
A poster indicating that someone was "Wanted Dead or Alive" simply meant that any law-abiding citizen could feel free to deal with that person without fear of prosecution.
It's interesting to note that there are some areas of U.S. law that haven't really changed much at all since then. If you have time, do some research into the fascinating hazy legal area in which legitimate bounty hunters (bail enforcement agents) still operate in the U.S. In many states, these agents -- who have no formal law enforcement authority whatsoever -- have powers that no police officer would ever be able to use, such as breaking and entering a person's home without a warrant, capturing that person without reading him his rights, etc.