moggle, in my experience the answer to your question depends on the size of your boobs AND the size of your baby. In the early days, yes, it's the boob closest to the bed that I think works best at least for my not huge boobs. And no, you shouldn't really let yourself sleep with the baby between you and your partner - but, if you can stay awake long enough, you can start on that boob and then switch to the safer side.
Also, no reason why you shouldn't still be in bed in the morning long after partner has got up - or, heck, why you shouldn't be in bed much earlier in the evening, too - so, you may well find you're not always with a partner in bed. I have found that it's quite easy to hold the baby to my tummy and roll over to the other side. Only when my baby was older (and when I think my boobs lost their balloon-like shape of the first postnatal months) did it work to feed on the boob further from the bed when on my side.
The other thing is that for a little baby, the previous suggestion about using a long hard pillow such as a yoga bolster or those long pregnancy pillows to prop the baby up on his or her side is really important - otherwise, a little baby will just roll back from her side onto her back. I think if you're doing this, it's not so bad to have the baby in between you and your partner (though not ideal for sleeping really), because the big bolster would still be between baby and daddy.
Hope that helps! I found it baffling at first, but breastfeeding in bed was my salvation - and I think I was a lot better rested than other mums, despite my non-sleeping baby.