A K-1 is the form a partnership sends each partner reporting that partner's share of income, loss, depreciation and a few other items. You don't file the partnership's return, you take the numbers off your K-1 and drop them into your own.
On the state side, owning a partnership interest in property in another state can create a filing obligation in that state, and this varies by state. Some states have no income tax at all, so nothing happens. Some require a nonresident return once your share of income crosses a threshold. Many partnerships file a composite return on behalf of nonresident investors or withhold at the state level so you don't have to file individually, and the offering documents or the sponsor's investor relations person will tell you which approach they use. Ask them before you wire, not in April. Your own CPA is the one to confirm what your specific filings are.
On cost, the common range people are quoted for adding a single K-1 with one nonresident state return is a couple of hundred dollars up to around $500, and it goes up with each additional state. A deal that owns property in four states can turn one investment into several filings.
The depreciation does pass through to you on the K-1 whether or not you operate anything, and in a syndication with a cost segregation study the first-year paper loss can be large. Whether you can use that loss against your other income depends on passive activity rules and your own situation, and unused losses generally carry forward to future years rather than disappearing. That's a conversation for a tax professional, because the answer is genuinely different for a full time real estate person than for someone with a W-2.
One practical thing: syndication K-1s often arrive later than individual tax deadlines, so filing an extension becomes routine once you own these.
