We gave up on cages many many years ago. Instead we use the weave technique. We take 10' rebar and inserted into the soil as far as we can, space them about 5' apart (we have beds about 20' long) plant tomatos between them (about 2' apart). As the plants grow we secure the twine to a post then weave the twine between the plants to hold them upright. As they grow we add another layer until eventually the plants are over 8' tall and still growing. We end up with a wall of tomatoes, off the ground, easy to see and pick.

Those cages usually are not worth the time, effort, and money. You'll need something stronger and taller to hold your tomato plants

Do I put them side by side touching? Or do I have to let there be space between the cages?

Most definitely need space between them. 3 feet is the average. Some of us use 4 feet.

The whole advantage to using cages is that you can do what they do naturally. Many branches will remain inside - depending on the diameter of the cage - many will poke out of the cages. Either way works. Early on in the season I do some tucking back in but as they reach full growth that is impossible.

If you browse through the discussions here on cages and spacing you'll find several pictures many of us have posted in the past that will be of help to you. These are a couple I have posted several times in the past. My cages are 30" diameter and set on 4 foot spacing.

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Same plants in August of that year.

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Dave

I use the Texas tomato cages I bought about ten years ago. Love them.
They are strong, strong, strong. I have extensions that make them over ten feet.
The tomatoes still grow over, but the spillage is minimal.

Every year, I move them further apart for ventilation. Keeps down diseases.
This year I have planted 4 feet apart.

We shall see.

digdirt and mockturtle, is that 4' apart on centers, or 4' from the left side of one cage to the right side of the next?