Hi, this is only my second post here.
My husband and I have had mutual dreams on many different occasions. The funniest one I can remember was several years ago, before we were married, we both had a dream that Jerry Seinfeld was in. Keep in mind, we never watched the show, we didn't talk about Jerry Seinfeld, and my husband doesn't even care for Jerry Seinfeld. It was so odd that we would both dream about him. We've had other mutual/shared dreams over the years.
The most recent one however was just several days ago. Basically we both had a dream that involved a turtle in an aquarium and trying to rescue the turtle. Keep in mind we do not have pet turtles, did not watch anything on t.v. about turtles, did not talk about or read about turtles.
In my dream, we were living in some sort of a large artist loft/co-op spare (don't actually live in one, own a house) and I had an art show going on that I was trying to get ready for. For some reason we had this very large aquarium with 3 medium sized/big pet turtles. One of the turtles escaped the aquarium because the lid of the aquarium was not totally covering the aquarium. My parents showed up to the art show and my dad was trying to help me rescue this turtle who was walking around on the floor. I didn't want anyone to step on or hurt the turtle. My dad and I struggled to put the turtle back in the aquarium and I suggested putting one of the window screens over the aquarium.
In his dream, again, somehow we had an aquarium with a pet turtle. I think in his dream it was one turtle and a small turtle. Our cats (or maybe it was just one of the cats) got the lid off of the aquarium and was trying to play with or even eat/kill the poor turtle. So, my husband was trying to rescue this turtle.
Obviously the exact details (such as size and number of turtles or how they got our of the aquarium or why we were trying to rescue a turtle) is not totally identical, but I would say there are certainly a lot of things in both dreams that do match up!
Now, one could argue I suppose that it may not be that odd or surprising that my husband and I sometimes have the same or very similar dreams. We share the same bed, it could be we talk in our sleep, and being married makes us close.
However, my husband is not the ONLY person I have had a mutual dream with. In high school, I had this awful physics teacher. It's a long story about this physics teacher and I won't bore with those details, but one night my dad and I both dreamed that this physics teacher was crying! Keep in mind that although my dad and I were sleeping in the same house, we were in completely different rooms when we were sleeping, so there is no way that something like talking in one's sleep would affect either of our dreams!
As for scientific proof, my husband is a very scientific/technical type of person. Despite that, he does believe in certain things that other scientific types might be skeptical about, such as ghosts. (Let me add that for most of my life, I never believed in ghosts and it wasn't until some other things happened that I was open to the possibility of ghosts or the supernatural or whatever you want to call it.) I have discussed how odd I think this mutual dreaming thing is, how odd it is that I seem to sense when someone has died, I tune into friends who live across the country, it just makes no scientific sense to me and I had to ask my husband, "Since you have a very scientific and logical brain, how is that that you can believe in this stuff too and how do you justify your beliefs scientifically?" My husband is an electronics/radio technician, so he deals with things like radio waves and radio frequencies. He basically explained his beliefs by saying that we all transmit and receive frequencies, just as radios transmit and receive frequencies. So, when he and I have the same dream, it probably just means our brains/souls (whatever) are sending and recieving wavelengths or frequencies. He believes we can send/transmit these brain frequencies way out into the universe perhaps and receive them from just as far away, which would explain things like tuning into your friend across the country while they tune into you, knowing when someone has died, etc.
So, sorry for the long post here, but yes, I do believe that mutual dreams happen and I have experienced them on many occasions, but never on purpose.