Playing In Your Own Territory Or Playing In Your Opponent's When You Have No Hope To Live
Though not strictly a mistake made by beginners, the most blatant attempts to live in your opponent's territory do come from beginners. Usually this involves outrageous second line plays facing fourth line territory boundaries and no aji in the region, nearing the end of the game. Usually attempts to make them stop (like passing, or yelling at them) fail, response most often see as "well, I gotta try." The most useful proverb with this is probably Six Die But Eight Live.
As for playing in your own territory, this usually comes from an inability to read out the results of a cut. Making solid connections where a tiger's mouth or keima connection is already present is the telltale sign here. Less commonly, it involves playing the 3-3 point to prevent the invasion when the local conditions already secure the corner.
Yes, playing in your own territory is OK, if it involves removing aji for profit - a 3-3 play in your own moyo may be a profit of 20 points. Doing so needlessly is the mistake.
See also speculative invasion.