>>How this gonna work though? Like you join that LS and have it's achievements?<<

You would get the LS achievements if they were done already. But would have no effect on your personal achieves.

>>Or like being in a party with everyone from the same LS to get an achievement? <<

example: defeat Ifrit 50 (insert number) times with full LS group. One non-member would not apply towards achieve.

>> And what activities would earn such an achievement?<<

examples:

Gathering: gather grade1 ltems 100,000 times. grade 2 items 250,000 times. grade 3 items 500,000 times. ect. All LS members with equiped shell will count towards achieve.

crafting: similar to above but for each crafting class.

battle: defeat 1 million enemies.

just for a few examples and im sure SE could dream up loads of them.