The process of getting into the other person's dream ('psychically projecting oneself into the dream', as it is referred to in the movie), complete with hypnagogic images and a feeling as if entering through a tunnel, is at first supported by some nondescript machinery, but later on both the positive and the negative hero are shown to be able to do it without any technical help, presumably by pure psychic ability. (Both the machine and the psychic stuff place the story firmly in the real of science-fiction; the former returns in Inception, the latter doesn't, which may be taken as a sign that even in the popular imagination there isn't so much room any more for mediums and the like, at least not where the story tries to establish the illusion to be based on sound science. Audiences, it seems, are more advanced nowadays in their views on what would count as a good candidate for plausible science and what wouldn't.)
But that process is based on the same idea in both movies: that the world of a dream can be shared between two or more dreamers (in Inception, the term is 'shared dreaming'), in a manner quite similar to that of a multi-player video game. One of the dreamers hosts the world, in that his sleeping mind creates the landscape and architecture, the interior, props and extras (people who don't really have a role, but just stand around to populate the scene). The other dreamers are guests in that world, but they interact with the host and his world quite as they would do in the real world. Let's speculate a little on what is involved in this.
In order to interact with another person's dream, the visitor to the host's dream world has at first to be able to perceive what's going on in that world. Let's call this intercepting the dream. In Dreamscape, the scientists who set up the shared dream scenarios indeed start out with experiments in which they instruct the subject to 'just observe' the dream. (Being a proper story hero, of course Dennis Quaid's character ignores these instructions at the first possible occasion.) Before any interaction takes place, the primary step is just to be able to see, hear, smell and feel what's going on, to get oriented in the dream world based on the sensory inputs it generates. It's that sensory input that is intercepted and interpreted, so that an additional perceiver can see, hear etc. exactly what the dreamer himself sees, hears etc. (Never mind the complications generated by different spatial locations and thus differences in perspective, distance, visual and acoustic obstacles and so on. Let's just assume the dream interception technology can handle these.)
There is a bit of a problem here already already with this idea. When you're dreaming, say, that you are in your house, your dream world is centered around what you see in your dream. You dream that you enter through the front door, and go up the stairs — fine, so the dream world comprises the front of the house, the front door, the hall, and the stairs. Since it is your house, it may also contain all the other rooms, the back of the house, the garden and so on. But then, it may as well not contain those. It's a dream, right? In a dream, all sorts of strange things may happen, and if you go round the house, you'll quite probably not be in the garden, but somewhere else entirely, perhaps on a mountain cliff, or in a tech museum. There's no way to know until you go there in your dream. But so far, the dreamer hasn't, and nobody knows what it looks like at the back of the house in that particular dream. And yet, in Dreamscape, the dreamer and the hero (Dennis Quaid) approach the dreamer's house and separate, the dreamer walking through the front door and the hero going round the house and entering through the back door. This scene already goes beyond mere interception of the dream: it's not just that the hero perceives the host's dream world. He also perceives aspects of that dream world he cannot possibly perceive, because they haven't been dreamed yet! In Inception, that problem is addressed by making it a rule that the dream's host is always one of the gang, and the dream world is designed in advance (by the architect) and taught to the dreamer. This little trick guarantees that the world in which all the participants in the dream move around is defined in advance, and properly dreamt.
 For instance in the positive hero (played by Dennis Quaid) who seems to be a player person in the beginning, driven by pure morality in the middle when he takes on the task of freeing a young boy and the president of the United States from their nightmares, and again not thinking too much about murdering a villain in his dreams simply because he might be dangerous in the future — all those different motives simply don't integrate into a plausible personality.