When you pass, your portion of the property goes to the devisees in your will or your heirs at law. However, in Texas, we have a Survivorship Deed which would allow the property to pass automatically upon the first to die. Maybe NY has a similar document. Check with a CPA on the taxes.
You should consult with a real estate attorney before you do anything as there are many considerations that come into play. There are also several ways of holding title and making sure your propterty transfers to the proper party upon a death. The size of your estate, whether you have other heirs and what you want to happen to your share of house upon death, your age and medical condition, medicaid and medicare considerations, long term care and tax consequences will determine the best way to handle this. For example, you may want to take the house out of your estate (which can be done by deed or trust) and leave it directly to your daughter and son in law or just your daughter, or you may want your share divided amongst other heirs (which you would do by Will). You could also decide to retain a life estate on the property and have the title in your daughter and son in law - giving them ownership and you the right to live there for life while still benefitting from any property tax exemptions you may be entitled to and keeping the property out of your estate.
As to who takes the deductions, that can be handled by agreement or in proportion to who pays and how much.
Speak with an attorney - this can be more complicated than you think.
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Tax issues should always be discussed with your accountant. As to survivorship after death, if the deed says that you co-own the property "as joint tenants with right of survivorship", then the property will pass to the remaining survivors upon the death of one. So if you die first, the property will pass automatically to your daughter and her husband. If her husband dies first, you will own the property with your daughter. If on the other hand the deed says you co-own as "tenants in common", then you own half and your daughter and her husband together own half so when you die, your half will pass to your heirs and the other half will continue to belong to your daughter and her husband. Your daughter and her husband will always own as "tenants by the entirety" as long as they are married so with regard to each other, they will always have survivor rights over the other's share of the home either way. Also, depending on who is going to live there, under some circumstances, it may be better to give them title to the property with you retaining a life estate (right to live there) but this you should discuss further with a local real estate attorney. Either way, consult your accountant as to tax issues and find a lawyer to help you out with the real estate sale/purchase.
Proceed with a great deal of caution before doing anything. Work with an Estate Planning attorney who will advise you not just on the mechanics of what you want to accomplish, but who will also question you privately about the nature of your relationship with your daughter, the relationship between your daughter and her husband., and other similar issues. There are distinct tax ramifications to transferring title now rather than doing it by will. There may be Medicaid benefits if you do a transfer sooner rather than later. If you do the transfer, you should have a clear written understanding before the transfer as to who will pay for what after the transfer and who will control what.
Good luck. This is a complicated situation.
My answer to your question is for general purposes only and does not establish an attorney-client relationship,