Can we interconnect with other people mentally during dreaming?
Short answer
Telepathy is pseudoscience. 

Background
I think you are referring to the phenomenon of telepathy, which can be defined as  


  [T]he purported transmission of information from one person to another
  without using any of our known sensory channels or physical
  interaction


and you ask whether telepathy is a scientifically established phenomenon.

Telepathy in general is dismissed as being pseudoscience (Jogalekar, 2013) as it has been debunked in various scientific studies (e.g. 
Ayeroff &amp; Abelson (1976)). 

Likewise, dream telepathy is considered pseudoscience. It was reportedly first documented by Freud in the 1940s. Freud founded many other pseudo-scientific concepts, including psychoanalysis (Jupp, 1977). Ullman &amp; Krippner from the dream laboratory in Maimonides Medical Center in New York City claimed they had proven the existence of dream telepathy in the 1960s and 70s (e.g. Krippner &amp; Ullman (1970). However, the studies were rightfully challenged. For example, many of their studies were based on a few or even on single subjects (e.g. Ullman &amp; Krippner (1966) and Krippner &amp; Ullman (1970)). Indeed, more recent studies have failed to replicate their findings (e.g.., Belvedere &amp; Foulkes, (1971) and Hearne (1989)).     

References
- Ayeroff &amp; Abelson, Personality and Social Psychology (1976); 34(2): 240-7
- Belvedere &amp; Foulkes, Perceptual and Motor Skills (1971); 33(3)
- Hearne, J Soc Psychical Res (1989); 55(814): 271-4
- Jogalekar, Sci Am (August 2013)
- Jupp, Philosophy (1977); 52(202): 441-53
- Krippner &amp; Ullman J Nervous &amp; Mental Disease (1970)
- Ullman &amp; Krippner Int J Neuropsych, 2(5): 420-37