My thoughts are that a number of fish eggs are initially "sticky" when released but at the moment of fertilization the exterior coating begins to harden and then the exposed portions of the eggs quickly lose their stickiness. In theory this would help keep fish eggs stuck in place they were deposited and at the same time make them more difficult to move them elsewhere. A bird would need to standing in the right spot at the right time in order to get fertilized eggs stuck to their feet, I find that highly unlikely.
With that said, some types of fish eggs such as grass carp the eggs do need to tumble in current for a while in order to become viable. Those types of eggs are likely not sticky at all.
