I didn't get to read through all 3 pages of responses so apologies if this has been said, but... I think your original post raises 2 different questions: 1) Is it normal for grandparents to want alone time with their grandkids; and 2) How should I decide when my child is ok with being left alone with certain people?
I know grandparents aren't "just anybody" and usually have a very special relationship with a child, but my experience/opinion is that I trust my child to tell me who she is comfortable with and who she isn't. In our case it happens that she's always been comfy with the grandparents at least in that they could hold her from the time she was very very small and she was ok. But we didn't choose to leave her alone with anyone until she was older.
But there are other people (extended family) who dd has ALWAYS cried around, from infancy through now (she's 2). A couple of those people have always fancied themselves godparents and have insisted from day 1 that we need to leave dd with them alone so dd can bond with them and be ok with them. My answer: ummm, NO! In my view, dd was clear that she wasn't ok with them and I felt like in time if she warmed up to them, then I might warm up to leaving her with them, but in the meantime it was NOT necessary (or even a good idea) to somehow "force" dd to adjust and accept being with them.
Again, I know grandparents are special, but if dd had not bonded with one of her grandparents and seemed to cry a lot around them, I'd have the same view: give it time, let's hang out a lot together, but I'm not leaving her alone with that GP until she seems comfortable".
Seems to me anyone who wants to force the issue with a child who otherwise isn't a really cranky kid but who cries around them, that's putting the adult's need for acceptance and quality baby-time over the child's preferences, and I don't think that's right. Grandparent or otherwise. I *do* think it's worth it to spend more time together as a group to facilitate the bonding and see if things change.... but I'm not gonna drop my kid off with anyone - even a GP - just for the goal of getting dd to bond better with thema nd get used to being with them.
By the way, in my case I fully and totally agree with dd's choices. Mostly she is wary of strangers and takes a little time to warm up to people she doesn't know. But once in awhile I'll introduce her to someone I already know I love and trust, and usually dd will instantly warm, sometimes even reaching out to be held by them right off the bat. Meanwhile the extended family above - they have a lot of drama in their home and I am fully ok with dd never being left with them alone.