Can we interconnect with other people mentally during dreaming? In some of my dreams, I have seen people that I know, and I ask them about it the day after and they say that they saw me too. My older sister said that she had exactly the same dream as one of her male friends.

Can we actually connect with other people mentally while sleeping?
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