It is a very big deal in the US but not here. In themed American puzzles the theme and its answers are the entertainment backbone, so plagiarism is definitely a bad thing. That said, the largest part of the solving experience is the clues and, while the various articles about Parker have casually mentioned plagiarised clues, no examples have been given, which kind of points towards the clues having a minor role (to the extent that plagiarism of them doesn't really bother people).
In the UK we don't plagiarise, especially in cryptic crosswords. Ideas are inadvertently copied from time to time, but it's inevitable. Setters are finding new ways of expressing things at a far greater rate than new words are being added to the language, so the 'pot' of originality is always decreasing. Repetition has to happen.
As for plagiarism, though, we actually go to opposite extremes. Myself and a number of colleagues use the search function of sites like FifteenSquared to look up answers we want to clue – not for inspiration but to avoid what's been used before.
Clues will get repeated though. They're like good jokes for which a new audience can always be found. Some setters (like Roger Squires) go as far as a manual filing system in which every clue is stored, along with its where/when publication details, so there is a record of prior use. Roger, I think, sets a minimum of 6 months before repetition, others go for shorter/longer. It depends on how many puzzles you have to produce weekly.
Entire crosswords get repeated, of course. That's just syndication – sadly, once setters have been paid their crosswords are owned by the client, so the setter gets nothing if the crossword is syndicated elsewhere.