I found that wet-nursing is a very emotional thing. Bonding with my newborn through nursing has always been one of the most special parts of having a baby. When my 4-week old developed thrush and I ended up with cracked, infected nipples because of it, I had to take about 3 or 4 days to rest my breasts and let them heal. I allowed my best friend at the time to wet nurse my son (since we were living in the same home). It was amazing and wonderful and I was so thankful - but it also hurt in a way much different than the thrush to see someone else feeding my beloved baby. Don't get me wrong, it was so awesome that I could do this and I was happy for my baby to be receiving breast milk... but I wanted to be the one nursing him, and I felt like a failure because my ""na-nas" were sick.
For all of that, when the thrush recurred a few weeks later, my friend and I were no longer getting along as well. I could not hand him over to her when she was refusing to speak to me, even though she later said that naturally her animosity would not have extended to the baby and she'd have been willing to wet-nurse again. It just hurt too much, though. So I gave him formula for a few days. He had some gas and he hated the silicone nipple, but luckily it was only for a couple of days. I went back to nursing just as soon as I could.
I don't think it's gross, weird or unnatural. I do think it's very emotional, though. It's not just a matter of handing your baby over (at least it wasn't to me). I was practically immobilized during that time - all I wanted to do was hold him and rock him because of the time that I could not spend nursing him. It takes a vast amount of trust to allow someone else to nurse your baby... again, this was just me.
Lil'Man is nearly 16 months old now, and I still nurse him.

He hasn't had to have formula since that awful experience over a year ago. I nurse on one side because the thrush eventually caused me to lose the milk on one side, but he thrives and loves his na-na.