I was never a fan of the flu shot but I had to fly from Cairns to Brisbane for chemo and so my oncologist insisted I have it. That was 3 years ago and I have had the shot every year since. I have never had any problems.
The flu vaccine is actually a killed vaccine and as such you can't catch anything from it. Your friends catching colds would probably have caught them anyway. I used to think I had had the flu lots until I actually got it once. It is very different to a cold and much more debilitating. I was flat on my back in bed with aches all over and fevers for 3 weeks and it took longer than that to recover fully.
Having said that, I actually had chemo twice due to a recurrence and the first time I didn't get the flu shot , or the flu. I guess I would base my decision on what the oncologist recommends and why. Also if you work with lots of people or are exposed to potentially sick people then I would definitely get it. It's a hard call, and one that people have debated about for years.
Good luck with the decision and also the rest of your chemo
Paula