Critical hits are still possible in conjunction with auto-crit skills (attacking out of stealth, Pinpoint Strike, Death Hex). All attack rolls which would normally be critical hits are converted into normal hits. All attack rolls which would normally be normal hits are converted into critical hits.
This ability behaves rather strangely, causing roughly 30% of the attack rolls to not connect at all regardless of attack rating, instead passing through the targeted enemy without reporting a *Miss* (occasionally hitting with normal damage but without triggering floating damage values). This occurs perhaps purposely as the ability's otherwise huge damage output increase without any penalties was too overpowered. This is persistent and various attack speed alterations (based on weapon type, Momentum, Swift Salve, Precise Striking, etc) do not seem to mitigate it in any way. Even with 30% chance to miss, this ability will increase the damage output by 40% which is very hard to beat even by a rogue who backstabs a lot.
One thing to note is that though attacks are made with both weapons, attack rolls are still separate for both hands, which means that there are cases when only one weapon attack hits the target. Compounded with the animation issue (i.e., the "miss" on the next attack), Dual Striking might end up encountering a net damage-per-second (DPS) loss. This becomes less of an issue once attack power reaches high levels.