Monte Carlo simulations have been used to study three different models for linear, titrating polyelectrolytes in a salt-free environment : (i) a rigid polymer with rigid bonds (rigid rod); (ii) a flexible polymer with rigid bonds; and (iii) a flexible polymer with flexible bonds. The use of a very efficient pivot algorithm has made it possible to simulate very long chains, with up to several thousand titrating groups. The results have been compared to a mean field approximation for a rigid rod and variational results emanating from a Flory type approach. It is found that the rigid rod mean field model gives a qualitatively correct description for the apparent dissociation constant for all three models. At room temperature, the energy contribution to the apparent dissociation constant often dominates over the entropic term, which partly explains the relative success of this approach. In the case of flexible bonds, both the conformational behavior and the behavior of the apparent dissociation constant are well described by a variational ansatz with a quadratic term, largely thanks to the harmonicity of the bonds themselves. The approach is less successful for rigid bonds, which becomes evident for highly charged chains where a harmonic entropy term is incorrect. This can be remedied by replacing it with an expression valid in the strong coupling regime. Empirical scaling expressions have also been found, primarily for the end-to-end distance.