Up until becoming pregnant I hate that we live about 1.5 hours from each set of parents. The biggest hassle with that is they are in opposite directions. Now that I am pregnant I am thankful for it. This means one grandma won't get an advantage of proximity.
We have already told our parents we will come to their houses once per month on the weekend (Saturday or Sunday) and then we get the other two weekends to our family. This has seemed to go over really well. Plus we will also do other family stuff with them but they will get one weekend per month that is theirs.
I haven't seen any competition per say. My MIL has made one side comment to the nature of has my mom left anything big available to get us or is she going to buy it all. I have specifically stayed out of this. I figure I cannot control how one spends their own money and it's not my place. If my MIL is so concerned about things not being left for her to purchase, our registry has been done for about 2 months now so she is free to shop as she pleases.
All I will say is we are going to make an effort to give each set of grandparents equal time at least until the "newness" wears off. Then I don't think it will be that big of a deal. I am concerned about my ILs as they do not have to "share" their other grand kids as my SO's brother has sole custody.
At least my MIL or my mom is not very vocal. They will just deal and stew. I am most concerned about the birth at the hospital over who will get to see/hold them first. At least with twins each grandma can hold a baby at the same time. I am actually more concerned someone other than my SO will try and hold them before I get to as I am having a c-section.