When I was growing up, I had to two sets of grandparents who I absolutely adored. Both sets had different backgrounds, and I enjoyed doing different things with them all. As far as I'm aware, there was never any competitiveness ... they just got on with being great grandparents, and my parents both got on well with their respective in-laws. This is what I've always hoped for in our family relationships.
We were already good friends with our DS's in-laws, even before they were married, spending lots of 'family time' together, before our GCs came along. But our DiL's mother has a tendency to be somewhat over-bearing & bossy, especially towards our DiL, who is actually rather intimidated by her own mother.
When our first GC came along, DiL's mum also became rather manipulative. DS, like most men, didn't want to rock the boat with his MiL, or cause any upset for our DiL. However, on occasions when we were present, it was noticed that her mum had upset/annoyed them sometimes. And, on one or two occasions, she extremely upset me. It was as though she was the only real grandparent! I'm the type of person who always likes to keep the peace (I guess that's where my DS gets it from!), but I got to the point when I just had to say something. I'm not particularly proud of that fact, and I do regret it now, as it did create quite an atmosphere for a while.
However, I do think that DiL's parents, and my DH & I, now have a mutual respect for each other as grandparents. It seemed to clear the air, so to speak, as there was a definite undercurrent previously, which didn't feel good. Very pleased to say that we do now all have a very good relationship, and we frequently socialise with the other GPs, even without our kids, or GCs. All we need now is for DiL to learn to stand up to her own mother.