From the royal website:
Privy Purse and Duchy of Lancaster
This is a historical term used to describe income from the Duchy of Lancaster, which is used to meet both official and private expenditure by The Queen.
The Duchy of Lancaster is a portfolio of land, property and assets held in trust for the Sovereign in his/her role as Sovereign. It is administered separately from the Crown Estates.
Its main purpose is to provide an independent source of income, and is used mainly to pay for official expenditure not met by the Civil List (primarily to meet expenses incurred by other members of the Royal Family).
http://www.royal.gov.uk/TheRoyalHousehold/Royalfinances/Sourcesoffunding/PrivyPurseandDuchyofLancaster.aspxUnder the 1337 charter, as confirmed by subsequent legislation, The Prince of Wales does not own the Duchy's capital assets, and is not entitled to the proceeds or profit on their sale, and only receives the annual income which they generate (which is voluntarily subject to income tax).
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He is in effect a trustee, and is not entitled to the proceeds of disposals of assets. The Prince must pass on the estate intact, so that it continues to provide an income from its assets for future Dukes of Cornwall.
http://www.royal.gov.uk/TheRoyalHousehold/Royalfinances/FinancialarrangmentsofThePrinceofWales.aspxAs for whether the palaces would be occupied by a president - not all of them, I'd say. Windsor Castle almost certainly wouldn't; or Kensington Palace, or St. James's Palace, or Clarence House. Holyroodhouse would be a subject of discussion between the Scots and the British government.