Our parents suddenly don't apply their rules in our upbringing to our children. On the contrary. How should we deal with this. Do we just let it be, or should we discuss the matter. I understand that a grandma role is different then a parent, but still it could undermine our authority or not?
I think it depends on the child's exposure level to the grandparents. In my experience, grandparents tend to spoil more when they see the child less. When they have a more active care-taking role, they usually stop spoiling.
Yes, you should always have open discussions with everyone that your child is close with on a regular basis on how you want your child treated, rewarded, and disciplined. This includes teachers, care takers, and your spouse / partner.
Find a balance...
It's all right to have a few special privileges that only apply at Grandma's house. That's a grandma's prerogative.
Being allowed to get away with bad behavior, eating only junk food, not getting enough sleep at night, or getting lavished with expensive new toys every visit, have to be nipped in the bud, fast.
Discipline only works if it is consistent. If a child can't do 'x' at mom's house and can at Grandma's, then it's a mom's house rule and not a generally applicable moral. If x="drink soda on the couch" or "watch TV before dinner", that's not a big deal. If x="wander off at a store" or "hit my sister" we have a real problem.
The sleep/food thing is, I hope, self-explanatory for health reasons.
As for the spoiling with gifts problem... this one caused a lot of conflict with my parents, but in the end they came to respect my wishes. My son, after an extended stay with my parents when he was a toddler, developed the expectation that going into any store necessitated buying him multiple treats, and that pouting and sulking and demanding would get him anything I was resistant to.
This made life miserable for me and for him. He was set in the idea that he "needed" every freaking thing that was in a commercial, and shopping with him was always a nightmare. It wasn't even that he liked all the trinkets so much...my behavior was just so outside his new expectations that it was extremely stressful.
Once my mom finally toned it down, she realized that yes, it made life more pleasant for everyone including her grandchild.
The best solution we've found yet is that the grands and great grands aren't allowed to skip the consequences either. No bungee-cord (grand)parenting allowed.
"If grandma and grandpa want to give you that giant plastic castle play set, then it can stay at their house so you can play with it when you visit".
"IF grandma is going to let you skip the nap and eat a pixy stick before dinner, then she can put you to bed tonight... maybe you could sleep over with them tonight."
Worked pretty well so far. Thankfully our parents are pretty sensible and caught on quickly with just a couple conversations like that.