Does each partner report their business expenses in the LLC partnership tax return or their personal return? Does the amount each partner reports as a business expense, affect the ownership in the partnership in any way?
The LLC will file its own business taxes which may or may not have business level income and expenses. At the end, the LLC will issue Schedule K-1 tax forms to the members, that based on their percentage ownership, will reflect the percentage share of the income/losses.
From an individual standpoint, the members need only worry about the K-1 form they receive. This has quite a few pass-through categories from the LLC, but the Income/Loss may be the only used one. The individual will likely include the K-1 by filing a Schedule-E along with their 1040 form. The 1040 Schedule-E has some ability to deduct expenses as an individual.
Generally it's best not to commingle expenses. Additional schedule-E expense reporting is generally for non-reimbursed, but related business expenses. If a member paid certain fees for the LLC, it is better for the LLC to reimburse him and then deduct the expense properly. Schedule-E is on a non-LLC, personal level.