Better to communicate with co-workers as equals/peers. Bragging is not necessary if people are aware of what everyone is working on and for what reasons, because the group's overall goal is what ultimately matters, not the status of each member.

There is a very good book on leadership and how to be a team member by the astronaut Chris Hadfield ("An Astronaut's Guide to Life on Earth"): https://www.amazon.com/dp/0316253030. It's a worthwhile read because if everyone tried to be as professional as an astronaut then most workplaces would be fantastic places to work. That book has generally good advice that is applicable in every workplace where teamwork is required to accomplish a bigger goal (which I figure is almost every workplace these days).