A partnership files an information return on IRS Form 1065. The partnership issues a Schedule K-1 to each partner, which reports their share of the partnership income. The K-1 becomes part of each partner's tax return.
There is a section in TurboTax for entering K-1's. Partners in a partnership are not paid a salary as employees.
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No, you do not issue yourself a 1099 from a partnership.
Your income from your partnership is reported on a K1 from the partnership that reports your allocation of the partnership's income and expenses.
The amounts from the K1 then get reported initially on page 2 of Schedule E on your individual Form 1040.