Gifting a Young Child - Etiquette Versus Expectation

It would depend on what the gift is. If it's appropriate for a toddler, then no, I wouldn't expect it to be run by me first.

I suspect there's more to this question though and that some context may reveal a bigger issue, such as someone with a pattern of giving gifts of which you don't approve, or giving gifts more often than you think is appropriate. But just a random, totally appropriate gift every now and again? Go ahead and make my child's day!

I'm not sure someone why someone would need to run a gift "by" me first, I mean people usually give toddlers toys or clothes, not power tools or cigarettes!
And if it WAS something I didn't really like or find appropriate for my child I would thank them graciously and get rid of it later.

Mostly my family runs things by each other (although I do not have toddlers anymore)...but I also let family know what the kids can/can't have whether it was gifts or food before and reminders as they would come for visits. But I don't expect them to we just do. There are times when my mom will know that we will say no but she does it anyway...that makes me upset (she and I talk about it) however, I do not deprive my kids of things just because grandma made a bad decision. i.e. she bought them tablets for Christmas. I really didn't want them to have them...but they do so we now have lots of rules around them, and have adapted to having tablets in our lives. They know the rules and for the most part have done well.

I do not have toddlers anymore. Yet, when it was time for their birthdays, I would tell family what they needed so I wouldn't get an overload of toys. I really loved it when my kids got bonds from immediate family members.

When my kids had a party, it was very small and I did not tell friends what to get unless they asked.

Some people like to buy a gift as a surprise. I always took that into account.

My friends knew that I liked educational toys/art and a gift made in the USA would be appreciated. They knew very well that I would not accept anything that had to do w/ violence...whatsoever.

As long as they are not giving my kid a PET then I could care less if someone gives my child a gift without running it by me.

I'd prefer that people run things by me, since my son also has food allergies, and since I'm pretty strict about things like videogames and electronic entertainment.

But if someone doesn't, I'm very much of the school that it's the thought that counts and that the proper attitude is gratitude, no matter what the gift is.

Totally depends. An appropriately chosen item (age appropriate, without any controversial possibilities?) would be fine. But, to say... promise a TV for their bedroom? Um.. NO.
Just a toy? Sure.. go ahead. Even a messy toy. Or a noisy toy. But a shootem up PS3 game rated M? ---you'd better check with me first.

If the gift is something that there is a lot of controversy around (toy guns for example---which I am fine with, but some aren't, and I wouldn't give one without asking the parent, if I didn't already know their position on them) then I would expect to be approached in advance. If it is a family member who knows our particular quirks or whatever, and wants to gift something they know is outside of what is generally acceptable in our household, then yes, I'd like them to ask me about it first. If it is something of a religious nature, then unless we share the exact same religious faith and practice, then I'd want you to check with me first.
A Disney movie, a package of playdough, an art set, a chalk board, dress up sets, etc...? Who cares...

I give gifts to children pretty frequently, but there are established parameters. I work with the children's ministry at the church and on their birthdays or as achievement recognition I may give them something. However, the parents are well aware that this takes place. If I give a child something at random, I would ask them to check with their mom to see if it is alright - even with my grands.

My mom and sister and close friends basically know what DD can/can't have. And little things, like giving DD a small toy, are usually OK. At her age, I'm usually there (me or DH) and can weigh in on the item given. If it's a safety concern, I take it til DD can have it or quietly get rid of it. For example, friends gave us a brass bell. Cool item, but not if the kid is prone to eating weird things. So they gave it TO ME for DD.

If people are consistently giving your child something he/she cannot have, then gently say, "Oh, thank you for thinking of DS/DD but I'm going to have to decline/hold it for now."

Yes, I felt people should run any gift by me first. We did not do character/tv based toys, toy guns, other violent toys, plastic toys made in China, toys likely to be swallowed or choked on, video games, videos or candy. Or particularly noisy toys.

I don't expect anyone to ask me if it is ok that they get my child something spacific. I think it is the thought that counts, and appriciate that they are thinking of my kid. For bday and Christmas, I do appriciate it when people call to ask what my child might want or need, but I do not expect them to ok it with me before they buy something. If my kids get something that they can't use, then I will thank the person giving the gift and either exchange it or donate it.

Mmm.... This is a little tough because there quite a few variables. It depends on who it is and for what occasion. For birthdays I usually request that no gifts be given. I may start doing the same for Christmas. I had someone give one of my children a Christmas gift that I would rather they not have had for various reasons, safety being number 1. Since I didn't know about it beforehand I didn't say anything. But I made sure to supervise them playing with it. Within a couple of days it got damaged (like I knew it would) and I quickly (and secretly happily) disposed of it.

For things like sweets, I find it annoying when someone whips them out in plain sight of my child and "asks" if it is okay for my child to have it. If I have reason to decline then I decline and I don't feel bad. In that case I would much rather I be asked in private.

It would only bother me if it wasn't age appropriate. Then I would ask that person to let me know what they plan on buying before they do so I could "approve" it.

Do you mean relatives or friends of the family or acquaintances?

I do not EXPECT, someone to run it by me first. But I have found, that it is relatives that will, more often, run it by me first. Just because they want to make sure of a gift idea first.
But I do NOT tell people, that they HAVE TO run it by me first.

You need to explain more. There are to many different variable to just hand out an answer.