Can only answer for myself - but yes if DS woke in the night he's always been welcome in my/our bed (was a lone parent when he was little). After a while he used to get very wriggly and I'd say to him "If you want to move around you have to go back to your bed" and he'd say "Okay" and either stop moving or go back to bed. I had a stairgate on the stairs. I did have to take him back the first couple of times and then I'd stay if I felt like it or sometimes I did just say I need to sleep now and go, he was usually fine with it.
He was in a full sized single bed at two though for this reason. I liked being able to settle him by lying down with him, it just made sense to me. I have never seen the appeal of a cot. (I do think bedside cots are brilliant). I'm not crass enough to call them "wooden baby cages" (which I've heard and always feel a bit 
) but they seem horribly impractical to me in the way that you have to lift the baby in and out and they can't get in and out on their own. I appreciate that's the whole point 
Since DS was about five he doesn't like sharing his bed space at all. He will occasionally come in for a cuddle in the night but he usually goes back to bed shortly afterwards. Lately he doesn't even do that, he just comes and wakes us up if he needs us or deals with whatever it is himself if he doesn't.
There is a theory that if you co sleep etc when they are little as much as they want they'll have had their fill and stop wanting it any more, whereas if you're pushing against it and trying to encourage them to sleep alone then they're more likely to want to sleep in with you because they haven't had it/done it, if that makes sense. I don't think it's possible to say whether it's true but I do wonder if that's where people get this idea that if you let them sleep with you for a little bit they'll never get out of your bed from. It doesn't seem to be a problem among, OK, for want of a better word, AP circles. Perhaps that's down to something else like being frightened to admit there is a problem in case people think you are uncaring or perhaps it's just different expectations to begin with and not a situation others would be happy with. Or maybe there is something in the theory - I don't really know.
Re babysitters, I barely ever left DS when he was little anyway because I was really clingy over him. Yep, I'm not doing great for this stereotype. When I did leave him it would be with family/someone I was close to so I felt like it wasn't a big ask. It's not like babysitting an older child anyway - you do expect to be a bit more hands on with a baby. I was also really laid back about routine so if DS didn't fall asleep until I got back or if he fell asleep downstairs on the sofa, it didn't matter to me. In actual fact most of the time he settled much easier for other people than he ever did for me, which I hear is very common. There were a couple of times when he just stayed up until I got home (or tried to). I always left breast milk or cow's milk and a bottle in case he wanted that kind of comfort to go off to sleep too.