I think it depends on how you define training. Depending on the definition I either never train or always train for my tours. By my definition I never really train.
On any given tour how important your condition going into it is will vary with the tour.
If you want to and can meander along doing 20 mile days on easy terrain taking it easy, it would seem that anyone with reasonable general fitness could manage with little or no preparation. Obviously having some time in the saddle would be a big plus, but that time in the saddle could certainly be very casual and not need to be anything that would have to be labeled as training. There would be a short time in the saddle spread throughout the day and there could be lots of breaks.
If you want/need to hit the road doing 60, 80, or 100 mile days and/or doing difficult terrain you probably need to be in better shape. Even then there is the question of whether the requisite preparation needs to be something that you call training. You likely want to have a good bit of time in the saddle, but that may just be your normal recreational riding for fun or a daily commute. Or maybe you are a runner and just supplement by spending a little time riding for fun on the weekend to get acclimated to the saddle. Is any of that really training? I guess it depends on your definition.
As far as riding into shape on tour. I have done that to a varying degree on different tours and think it is a good idea to do it but not rely on it to heavily. By that I mean show up in at least somewhat decent shape, but take it a just little easy the first week or two on a long tour. Especially for that first two weeks ride a pace and distance that you can handle without burning yourself out. Pushing hard enough that you need a rest day is a bad sign especially in the first two weeks IMO. In fact I think that needing a rest day is a sign you went way too deep. Note that I am not saying that you shouldn't take a rest day to enjoy it, that it shouldn't be because you overdid the riding. My personal preference is most often to save the rest days to do fun active off bike stuff (hiking, tourist stuff, whitewater rafting, peak bagging...) and even then it is most often half days with some riding. I have only rarely taken a complete zero day with no riding at all, but fairly often take a short day to rest.
BTW, I think all that stuff about rest days and pushing to hard applies whether riding into shape or not.