All my daughters' shoes are hand-me-downs except for her athletic shoes. I figure she isn't spending enough time in her other shoes to cause any serious damage.
Let me elaborate.
I had an aunt that bought my sister and I special shoes designed to help children walk correctly and healthfully and gave those shoes to my parents. She continued to "keep us in these shoes" by purchasing the next size every six months or so until she passed away.
By the time I was a year old, my parents had learned not to use the shoes, despite her purchase of them. We constantly had ingrown toenails. The ingrown toenail thing actually runs in the family and was not caused by the shoes, but because of the problem, Mom had consulted a physician on the matter. The doctor said, "The best thing for babies are bare feet." I still have really healthy arches and have feet "ballerinas would kill for" because I have such a nice arch. This article at unshod.org, and the doctors quoted in it, gives more information about why this statement is true. Thinking about things this way, I would guess that if the shoes are to be worn a lot, more damage is likely done by shoes that don't fit quite right or are worn out unevenly than newer shoes. But knowing my daughter spends so little time in the shoes, I'm not afraid to take hand-me-downs from family members and let my daughter wear them.
I want my daughter to have healthy, new, correctly-fitting, supportive cross-trainers if she's going to be doing something like hiking, playing tennis or basketball or some other sporty activity. We tend to buy one pair of her own cross-trainers as needed as she grows.
For everyday activities around the house, she goes barefoot. If we go to run errands, she is often in flip-flops, sandals or whatever she happens to grab (or, in winter, knock-off Uggs!). Since she never spends a lot of time in these shoes, as long as I know they're clean, I don't worry about it too much.
It seems natural fibers and materials are less likely to carry these kinds of diseases (or at least hang onto them) based on a tiny bit said at the very end of a health 24 article. And based on one of the links provided in unforgettableid's answer, socks also create a pretty good barrier in preventing spread of such diseases. I have always just washed hand-me-down shoes using the hot water cycle in my dishwasher. (Weird? Yes. But it works.) My daughter has never had any such infection. Nor have I, come to think of it, despite serious blisters and all kinds of nastiness from dancing en pointe through college.
Having feet that are moist a lot and for long periods increases the likelihood of infection. Perhaps the fact that my family wears socks when wearing closed shoes, and goes barefoot in the house, means that our feet just don't get sweaty and damp enough to encourage the growth of such diseases. This is another reason I'd personally recommend the barefoot life as much as possible.