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English: We have three other colleagues who work with us on the team.
German: Wir haben noch drei andere Kollegen, die mit uns in einem Team arbeiten.
My translation for this sentence into German would be exactly the same except without noch.
Is it needed for the sentence to work?
How does it change the sentence?
I would suggest, that noch emphasizes a further in this case. So, translating each of both sentences seperately I'd come up with
English: We have three other colleagues who work with us on the team.
German: Wir haben
nochdrei andere Kollegen, die mit uns in einem Team arbeiten.
Respectively
German: Wir haben noch drei andere Kollegen, die mit uns in einem Team arbeiten.
English: We have three further colleagues who work with us on the team.
The second sentence with the noch strongly suggests that there is already a known amount of colleagues on the team and the three mentioned in the sentence are being added to that amount.
No, the word noch is not needed here for the sentence to be grammatically correct.
However, I would still put it in this case. The word noch effectively joins the first group of people ("we/wir") with the second group of people ("drei andere Kollegen"), which is important because in the last part of the sentence you speak about the team consisting of "wir + drei Kollegen".
An example where I would not necessarily use the word noch:
Wir haben drei Kollegen, die immer krank sind.
Because in this case the colleagues are presumably different to the "we" group of people.
So the word "noch" is not needed since without it it's a correct sentence. But: The word "noch" is not superfluos as it gives a slightly other meaning, depending on the context of the sentence.
German: Wir haben noch drei andere Kollegen, die mit uns in einem Team arbeiten. English: We have three other colleagues left who work in a team with us.
Which means, except us, we've had more, had lost some and have left over three.
Another meaning could be:
English: We have three more colleagues who work in a team with us.
"... uns in einem Team arbeiten," or "...with us on the team" refers to a "core" or "understood" group.
"Wir haben noch drei andere Kollegen" means we have three additional colleagues (who are not part of the core group). The "noch" is for clarity and emphasis.
If you omitted the "noch," it would not be clear whether the three colleagues were members of the core group or not.
