Two users at the two ends of a bidirectional channel wish to estimate the common state of the channel. The problem is usually treated as two separate one-way channel estimation problems: User 1 sends deterministic pilots to assist user 2 in estimating the channel, and vice versa. This paper questions whether such separation is optimal. In other words, is it beneficial to let a user choose pilots that adapt to what the user has learned about the channel? Two concrete models are studied and it is found that using adaptive pilots often improves the channel estimate. In the special case of a Gaussian channel with colored additive interference, an iterative bidirectional estimation scheme is proposed, which achieves significantly better performance than separate one-way estimation.