When you activate a lev in a party, they do get shared with others in the party. It is shared in a way that you all get to participate and complete the objectives. If for example, multiple people have the same lev, you can lev link them to gain a bonus to sp/exp when killing the lev mobs. To lev link, have 1 person activate the lev in the normal fashion. Have each additional player click the aetheryte and choose the "lev link" option. This will then allow players to complete the lev at the same time. You could, if you choose to, activate one then finish it, then have another player activate theirs. You'll just end up doing the same lev twice. Lev linking allows multiple players to complete the same lev in the same 30 minutes. It saves loads of time. The person/people who activate/link the lev will receive the reward quoted when accepting the lev. Those who didn't will receive a 1 or 2k for tagging along.
If you and other players have different levs, I suggest you link the ones you have in common and then do each of the others together as a party. This way you get the sp you wouldn't otherwise get from levs that aren't in your journal. For example, I have 1 lev and another friend has a different one. I could do my 1 lev by myself and get the sp for it, or I could team up and do my lev (basically the same sp amount gained) and then also get the sp for helping my friend out with his. We both basically get sp for doing 2 levs instead of just the 1 that is in each of our journals.
I apologize if this is very long winded. I just wanted to make sure I explained it fully. You can look at the adventurer's guide in game, or hit me up with other questions. My best advice for you in this game is to find yourself an active linkshell to join. You'll not only find others who enjoy playing the game, but also a group of people who will be willing to party up and help along the way. Happy adventuring!