Taxes are being paid on this, whether the property is in the trust OR the beneficiary's names. The difference is that trusts and estates have a steeper tax bracket than individuals. So it is likely that your tax burden would decrease slightly, if the property was transferred to the beneficiaries. The beneficiaries do not need to "assume" the debt. The property is held, subject to the debt, but the individual beneficiaries are not personally liable. Of course, the flip side is that, if the debt is not paid, the property can be foreclosed. Since it sounds like the income from the rental is more than covering the expenses, I do not know why you would not want to either keep these properties or sell them for a profit.
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It might help to consult with an attorney who is licensed to practice in each of the states in which there is property. A local attorney would be able to advise you on the ins and outs of the local tax structures.
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Whether or not the properties can be transferred to the beneficiaries depends on the terms of the trust. You may also wish to consult an accountant or CPA to learn more about the tax issues. I would start with an attorney in the state where the trust was written (or the state who's laws govern it) to get an interpretation of the trust terms.
This answer provides general legal information and should not be construed as legal advice to be applied to any specific factual situation. It is not intended to create and does not create an attorney-client relationship. The attorney writing this post is licensed in Texas and Washington only and the laws of your jurisdiction may differ.