First the obvious and worst way to try ...
In reality this would require extensive individual calibration of a complex system to another person's brain activity. This is the killer for Inception-like invasive dream sharing as you'll not get access to the person for long enough to do that. The calibration process would need to be on the conscious brain (or at least a semi-sedated person) and would require a lot of monitoring and pattern matching by a computer to work out what areas of the brain are active when e.g. an image of a chicken is shown (think how many variations are required - millions on millions).
And you can't stop at just images - you need to get sound responses, smell and taste responses, touch responses.
This process is fine on the invaders, because you can take time to do the calibration. But on the invaded mind you have no time for what could be months of calibration.
And human minds change. You have to keep adjusting the calibrations.
It's extremely doubtful you could ever do this in reality simply because no more-or-less passive scanner would be able to resolve the activity in enough detail, but assuming I'm wrong you'd need ...
Something like a variant on an MRI scanner
A lot of computing power - we're talking about something on the order of multiple top supercomputers.
The computer reads the data and has to reconstruct a virtual representation in real time.
It has to do this for two or more people at the same time.
Based on what all of them are generating by was of dream data, it has to construct a model (in real time !) of that weird dream scape and try to merge the scenes and to e.g. translate positions (which may be imagined differently in each person) and communications into a mutual form that can be sent back.
And the big problem : making fine tuned adjustments in real time to the operation of what could be individual cells in the brain of each person.
I hate to use the expression "impossible", but this is high on my list of things that sound far too hard to bother trying.
What you'd actually do ...
Sod their dreams. You hypnotize them (or some more sophisticated variant that leaves them in that kind of state). Hook them up to a virtual world controlled by the computer which they see through VR systems (and feel to some extent).
Comms from the invaders and scenarios from the computer can now be constructed and controlled in real time with relative ease. How do the invaders speak to the invaded mind ? They speak into the microphone.
Because you don't have to control a brain or decode it's activity, you have removed a huge stumbling block.
The computer doesn't have to be super anything, just be a comprehensive VR system feeding conventional inputs.
The human brain will do the rest for you.
From the point of view of the invaded semi-hypnotized person, they'll believe they're in a dream (or whatever) and their own minds will massage the input it gets to match that. This is what people do all the time : map what they get as inputs to a model in a way that matches what they expect to see and hear. It's why e.g. optical illusions work : your brain takes what it sees and maps it to what it expects to see.