In Oregon, pretty much any law-abiding adult can get a license to carry concealed. In Washington State, this has long been the rule. And in Vermont, people have pretty much always been able to carry concealed guns even without a license. There’s nothing specifically Texas / wild West about this rule; indeed, Texas law forbade concealed carry (though it allowed open carry) until the mid-1990s.
Today, basically 42 states allow pretty much any law-abiding adult to carry concealed guns in public — either requiring a broadly available license, or, in a few states, not requiring a license at all. I’ve heard some uncertainty about one or two of these, such as Connecticut, but the big picture is that right-to-carry is as much Washington, Oregon, Maine, New Hampshire, Indiana, and Pennsylvania as it is Texas or Mississippi. Maybe it will soon also be California, and the San Francisco-Portland-Seattle connection will be still stronger.
In the meantime, here’s a cool animation showing the transition from nine right-to-carry states in 1986 to 42 today; I can’t vouch for every detail, but the big picture is correct. (“Unrestricted” means carrying concealed is generally allowed, even without a license. “Shall-issue” means that pretty much any law-abiding adult can get a concealed carry license. “May-issue” means that it’s up to local sheriffs or police chiefs to decide who can get a license. “No-issue” means that people can’t carry concealed, period, unless they are police or some similarly exempt group.)