Yes I think healthy ovaries take it in turns but not always. I had a scan last month and they said my right ovary was about to release an egg. I have just had a positive Ovulation stick and a peak on my monitor and I have pains in my left side so I am guessing I am ovulating from the left one now.
The internet says
"Yes, pregnancy is possible with only one fallopian tube as long as you have one or both ovaries and still ovulate.
Typically, a single egg is released each month by one of your ovaries. The egg travels down a fallopian tube, where a sperm may fertilize it. The fertilized egg then continues on to the uterus. You need only one fallopian tube for this to happen.
The egg released from the ovary usually enters the fallopian tube adjacent to that ovary. Although evidence suggests that an egg released from one ovary can be transported by the fallopian tube on the opposite side, this is rare. Ovulation tends to alternate ovaries, so it's likely that with only one tube, eggs from the opposite ovary never have a chance to be fertilized.
Also, fertility may be reduced if a fallopian tube was removed due to a tubal pregnancy or infection. In such cases, the remaining tube may not be normal. If you have only one ovary and it's on the opposite side of your only fallopian tube, you're at increased risk of tubal pregnancy."