Okay, full disclosure: I have an almost 11 year old son. He's not even remotely interested in spending time with girls. Yet.

So, my advice is only based on theory. And I probably have a "do as I say and not as I did" approach.
I think 13 is entirely too young to have a "girlfriend" or "boyfriend". Yes, I realize that in most cases it's like you guys described. It's innocent puppy love at school. Not real dating. Kids need to test the waters and a little innocent "I heart so and so" drawn on their notebooks is fine. The problem is that kids today are growing up much faster. The age of the technology has brought the internet, cell phones and text messaging into the mix.
I have a problem with kids being on myspace and other social networking sites. Here's why (warning sorta long rant ahead):
My nephew (who is almost 14) has a "girlfriend". I put that in quotation marks because he's just a kid and they only talk online and on the phone. They don't even go to the same school. Anyway, when I was home for Christmas, I took him shopping to pick out a modest gift. He got her a cute little necklace from Kohl's. My brother and I drove him to her house and they did the awkard present exchange and it was really sweet and cute. They were both mortified because we were there.
When we got home, my nephew showed me her MySpace page. She's ELEVEN!!!!!!!!!!!!!!! Yes, that's right. ELEVEN YEARS OLD. JUST TURNED ELEVEN BEFORE CHRISTMAS.
My brother (his dad) and my mom (his grandmother) had no idea!!! She goes to a different school but lives nearby. Well duh! Of course she goes to a different school! He's in Jr High and she's not!!!!
I was shocked because she looked 14-15. She was "developed", wore lots of makeup, and seemed pretty mature for her age. My nephew told me her date of birth and I did the math. She's really 11!
My brother and I were in agreement that he needed to stop seeing her. Apparently her parents don't care but we certainly do!!! Granted, they are just kids and have never had any alone time, but still!!! ELEVEN years old people!
In my nephew's case, there was a serious lack of communication between his mom (my ex sister in law), my brother & our mom. They all live within minutes of each other and my nephew spends time with all of them. My brother really thought that my ex-SIL had "approved" of this relationship and that the little girl was at least the same age as my nephew. My mom turned pale when I told her how old she really was. She had seen her MySpace page a few times because her computer is in the kitchen and he's only allowed to surf the net from there. They watch over him and never thought to ask "how old is this girl?" because of how mature she looked! And, my brother had met her a few times in person and he didn't question it either. He was just as shocked as the rest of us.
My SIL leaves a lot to be desired as a mother and I can totally see her not even thinking to ask. I'm also a bit miffed that she even allowed a "girlfriend" but my mom just rolled her eyes and said, "Well, they aren't really 'dating', it's just puppy love over the phone/texting/IM's. My brother also had the attitude that it was harmless since they never spent any time alone - he was always with him whenever he went to see her. What's more jaw-dropping is that HER parents allowed all of this!!!!!!!
Needless to say, I too have a lot to learn as a parent of a boy about to discover girls. I honestly don't know what our policy will be.