It would be interesting to know how Moodle quiz use divides between formative assessment (assessment for learning) and summative assessment (for grading/ranking). Online assessment is never any good for high stakes summative assessment surely, unless you invigilate to eliminate all the various versions of "phone a friend". I completely support the idealist devil that Tim finds in himself! But Moodle sometimes does seem a bit unfriendly for formative use, which many institutions are really trying to encourage (more self- and peer-assessment, more collaboration, more feedback, fewer tests and exams). It could be (I hope) that my impression is out of date though, because my experience and discussions were a year or so ago.
Issues I remember encountering were that it was cumbersome to give immediate feedback each time a student answered a Q, that Moodle didn't easily enable students to enter comments linked to a Q that would be easily visible to other students in the same context, to generate interaction and constructive improvement; you couldn't use runtime randomisation of variables in math Qs, and that once use of a quiz had started it wasn't possible to correct errors or make improvements in it. The way things worked was good for summative assessment but the missing features would be really valuable for stimulating effective learning. In this context, as Tim says, it should be easy to persuade students that cheating is really stupid - if you genuinely reassure them that you aren't going to use their formative marks in any way that might incline them to cheat. One in-between strategy I've used is to allow them to redo math tests (with randomisations) as often as they want, with the requirement that they must eventually reach a pass mark; they could cheat, but probably seldom do, because it's pretty obvious that by behaving properly their errors don't count against them, and they will learn something they need. There's no point in cheating when practicing your tennis strokes against a wall!