This came out of a conversation Shinji's player and I had, which in turn came out of the fact that we're having quite a bit of interaction between the Universal Century (Gundam) and Evangelion casts. (Yulia and Avel's romance, plus Amuro, Shinji, Kaworu, and Misato all being teammates, among other things.)
So, here's the basics:
Newtypes and Children, while both being a form of psychics, operate on different wavelengths, so to speak. Newtype abilities originate in the upper brain functions, while Children's abilities operate on a more primal level. (Or as Shinji put it: the power of EMO!)
Where there is some small overlap between the two, any sort of psychic communication between them is considerably more difficult and takes a great deal more time and effort than it does with one of their own. Frankly, in most cases it'd just be easier to just talk to each other.
Newtypes may be able to communicate with an Eva on some level, but wouldn't be able to synch with one enough to get the Eva to operate. Conversely, a Child would quickly burn out a Psycommu system, if they could get it to work at all.
Angels (particularly the ones towards the end of the series) have also demonstrated some psychic abilities. Since their thought process by all appearances (with the possible exception of Kaworu) is very different from human, chances are that both Newtypes and Children are at best going to have headaches and at worst need a padded cell after communicating with one.