Suppose we have a prior distribution of $$\pi_1(\theta)=k\theta^{k-1},$$ and we have another of $$\pi_2(\theta)=k(1-\theta)^{k-1}.$$ Intuitively, what do the two different distributions mean / represent?
I understand that the summarise the beliefs of the tester before and after the test, but I don't understand specifically what these ones mean. Are they not both very much the same, since if $\theta\in(0,1)$, then so is $(1-\theta)$, and the only "difference" is that one is analogous to probability of throwing heads, while the other is analogous to throwing tails - by symmetry aren't both these prior distributions "the same"?
I'd like to understand what $k$ and $\theta$ represent.
Note: both have $\theta\in(0,1)$.
The second function you mention leaves no doubt that you're talking about probability density functions and not about cumulative probability distribution functions.
The first one concentrates probability near $1$ and puts little near $0$. The pair does have just the kind of symmetry that you mention.
The first one is the probability density of the maximum of $k$ independent observations that are uniformly distributed in the interval from $0$ to $1$.
The second one is the probability density of the minimum of those observations.