Dreamspace was a place with the conglomeration of all of an individual's possible dreams. All individuals were constantly dreaming every possible dream, even while awake. So, as a part of the brain stirred that brew, this resulted in the individual's dreamspace. Which images and thoughts one chose to remember was based on their own wishes and anxieties.
Among an individual's own dreams, the collective Slayer memories and prophecies were shared in this place as well. Notably, the Slayer Melaka Fray didn't have them in her dreamspace, as her twin brother, Harth, received these visions instead, which she only received after his final death. Buffy Summers had various prophetic dreams as a Slayer, such as her incoming dangers after moving to Sunnydale, Jennifer Calendar having a secret related to Angel's lost soul, the arrival of the Gentlemen, Potential Slayers being hunted by the Harbingers of Death, among others.
The Slayer Sineya, as the Primeval One, would manifest her spirit in dreams in order to communicate urgent but cryptic matters to current Slayers. She had done so with Buffy to express her dissatisfaction with her efforts against the First Evil, as well as reveal Buffy's false (robotic) constitution. Exceptionally, Sineya once manifested in the dreams of Buffy and her friends — Willow Rosenberg, Rupert Giles, Xander Harris —, and attempt to kill them in punishment for invoking her spirit in an enjoining spell.
Living Slayers were also able to share dreams, as well as cryptic messages about the future, as Faith Lehane and Buffy twice talked about an upcoming sister, despite neither of them actually knowing about her unexpected arrival. This ability of communicating through dreams extended to the thousands of coexisting Slayer as well. In a particular event, Buffy and the vampire Angel were also able to share a dream in which they had sex, manipulated by the First Evil in attempt to convince them to physically do so.
Through magic, it was also possible for one to access the dreamspace of another. Under the influence of the mystical drug Orpheus, Faith walked along Angelus through various of his memories. While Buffy was trapped within her own, Ethan Rayne was able to manifest himself in this place to advise her, even seeing some of her dreams. Similarly, Willow invaded Amy Madison's dreamspace and saw her nightmares, then used her fear of her mother Catherine against her.
Although not directly mentioned as such, the astral projection into one's mind to see memories or dreams involved the access to one's dreamspace. It was the case when Willow tried to reach a catatonic Buffy, using a spell that allowed her into the mind of the Slayer, and was able to see and interact with her significant memories and thoughts. She would also access one of Buffy's Slayer dreams in order to help her understand it, only to discover the cryptic message was also intended to her; at the time, a fairy nicknamed Tink was accessing the same dream, initially intending to transform it in a nightmare.
In another example, Buffy's astral form was sent into Spike's mind, and was able to see his memories, dreams, and impressions. In his mind, she also found a trail of shadow into Archaeus' bloodline, following it through Drusilla, Angel, Darla, the Master, and even be physically attacked by Archaeus himself. The demon lord himself had been manifesting in Angel and Spike's dreams; through their bloodline connection, the vampires dreamed of Archaeus killings as if they were their own.