The prior is a uniform distribution, or actually a discrete approximation of a uniform distribution.
Each person in the room gets a difference value of p and a degree of confidence.
If we get heads, each person updates with
and if we get tails, the update is
After each flip we can look at the location and shape of the distribution.
With a large number of flips, the distribution converges on the actual value of p.
1) It's ok to start with an unnormalized prior, but we do need the hypotheses to be ME and CE (mutually exclusive and collectively exhaustive).
2) You can normalize the posterior after each flip, or leave it until the end. Same answer either way.
3) The results depend on the prior, so in that sense it is subjective. But we can often use context to make justified decisions about the prior.
4) With enough data, people with different priors converge, unless the priors are "immune to data."