Do They Know? by MHKM

So after last night's episode, finding out Saywer is now a cop, I got to thinking about LA X and the implications from Recon.

Does this then mean that the Losties in the sideways world are aware of the on-island time line, at least to some extent?
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I know we have been debating what the moments of pause mean among the sideways Losties. Are they recalling experiences from the on-island time line, have they known since we first saw them on the uncrashed version of flight 815, or were they just passing gas?

I think now knowing Sawyer is a cop in this world coupled with him obviously seeing Kate in the elevator in LA X with cuffs on and doing nothing, actually assisting slightly in her getaway, does that infer that they all know or are aware of the on-island events, at least the ones that were touched by Jacob in the on-island time line?

or

Did Kate flip his switch that well at first sight, as she did in the on-island time line, to get him to do anything, even breaking the law for a complete stranger in cuffs running from the cops.

This seems a bit far fetched, I think THEY KNOW!

p.s. In LA X, didn't it seem Sawyer had a devilish look on his face when he ran into Kate in the elevator, like he was enjoying it somewhat, being all cheeky with her, seeing her in cuffs on the run, laughing about it a bit on the inside..."Better get going Freckles, wouldn't want to get caught again..." or something like that...

Maybe I'm stretching here on the last part but I thought the interaction was odd...