(i.e, legally free to marry, consider themselves married, use the same last name, own or rent property together as spouses, stuff like that.)
In states that recognize common-law marriage, it is JUST AS VALID as a ceremonial marriage, and in fact, you must obtain a formal divorce to end it. I've run into a lot of people who didn't know that.
Back in about 1977, my stepmonster was exed for living with my dad unmarried. He even left my mom to move in with her, so he was still married for part of that time.
Later, she got him to take the lessons and he was baptized in 1978. She had to go through the repentance process (which probably involved converting my dad) and then they waited a year so he could re-baptize her. They did their first temple trip shortly thereafter.