I've been vegetarian for over thirty years and I can handle meat, cook with meat, sit next to people eating meat with no problems. I don't particularly enjoy preparing meat, but I'm the only vegetarian in the family so I do it every now and again. I've also been accidentally fed small amounts of meat on two occasions.
Most vegetarians that I know are vegetarian for ethical reasons or due to a simple dislike of meat, and in the absence of real health risks attached to contamination (as in the case of allergies) I think it's probably sensible to develop a fairly robust approach to the whole thing.
Mistakes are occasionally made, and there will be all sorts of situations where people might serve food without it occurring to them to avoid using different utensils, and even more situations where you have absolutely no idea how your food has been handled. If you are vegetarian due to animal welfare concerns, then the most important thing is that you are not adding to the overall demand for meat. If it's about personal taste, the most important thing is presumably that you don't actually consume something you don't like.
It's possible to have strong ethical views, and also a pragmatic approach to things. You can choose to get worked up about the idea of something touching meat and then touching your food, or you can try to avoid it but recognise that the occasional error doesn't actually impact on the real core of your values, particularly if actual consumption of meat isn't involved.
As a PP mentioned, vegetarianism is something of a halfway-house between eating meat and eating no animal products at all. If you have such very strong ethical feelings about meat that you are distressed by the idea of utensils touching meat, then it's odd not to go the whole way and become a vegan. I'm vegetarian, not vegan, and I can accept that it's actually a slightly mixed ethical stance to take. As it happens, I turned vegetarian because I don't like meat or fish, although if I had to make the choice again now, there would probably be a much stronger ethical element to the decision, as I know more about the meat industry/environmental issues than I did back then.
Having said all that, I would avoid eating vegetarian food from a meaty barbecue, as it's pretty much impossible for it not to finish up covered in meat fat and therefore, to me, inedible.