Apologies for the very late reply.
Boast benefits are immediate.
The out-boasting is meant to be more collaborative than competitive. Players could game the system, but I would just ask them not to. I find writing rules "defensively" gives very bad results.
Being able to out-boast is something that NPCs can do too. The idea being you can egg someone on to taking on a too-large Boast, and getting themselves killed.