My DH's proposal was not a normal one....but shopping together to pick the diamond was fun. Here's my story.....a rather unusual one.
DH and I talked about getting engaged 6 months prior to the proposal. He told me his plan was to surprise me with the ring. Being picky about diamonds, I asked what he had in mind. He said the cheaper and clouder the better.

I think he was kidding though, but I kept hinting to him my ideal diamond. Months gone by and still no ring. Then out of the blue one night after dinner, he got down to his knee and proposed to me with a cheap ring he bought from the candy machine. I was totally in shock.

Ok, at that point, I thought he was such a cheapskate and probably the cheapest guy in the world, but I said yes anyway. Bottom line was I really love him and the ring didn't matter....but can't wonder why he did that. I hated the ring and can't see myself wearing it for the rest of my life. The cheap bendable metal will eventually give out. Actually, I was too embarrassed to show it off, fearing everyone would laugh at me. Two weekends later, he asked me to go with him to the jewelers to fix his father's watch. There I looked at the rings on display and fell in love with a ruby ring with diamonds, but the setting wasn't to my liking. The jeweler said he will find a better ruby. Then DH said surprise! Pick whatever you want and that will be your engagement ring. Whoo! Now that's my guy.

We spent the next several days finding my ideal ruby and loose diamonds for the setting. I was very specific as to want I wanted. When we finally found the one, his parents and my parents had an issue of it not being the traditional diamond ring. So, we ended up looking for diamonds only and found a beautiful colorless, round diamond for my er and as a bonus, he insisted that we get another colorless round diamond for a pendant.
So, I got a set for my engagement.... the ring is to tell the world that I belong to him and the pendant to tell the world that he will always be in my heart.
So, back to your question, shopping together is common and a great experience. You don't want to end up with the ring you're not 100% crazy about.