That's not true.
They are responsible for some of their personal expenses. However, this is an insignificant in terms of the cost - and may end up being nothing. For example, they are responsible for their personal lodging and the cost of a first class ticket. They are supposed to pay for tickets to events, admissions, personal food and incidentals. However, presidential lodging is often donated, with the owner's compensation being in the form of 'the president stayed here!'... plus reimbursement for anything used by security and staff. The government pays for modifications to the property (communications and security), use of facilities by security and staff, housing/per-diem for security and staff, transportation, the actual cost for Air-force One (and the backup) other aircraft, vehicles, etc.
But there's also an exception - official business included and the tab goes to the government. Fly to Hawaii for two weeks... spend an hour giving a speech at an elementary school (take the wife and kids!) and the flight is covered. Meet with some local officials, shake some hands at a military base, and those days become work. Since the President is never really away from work, and can have staff arrange little meetings, it's not hard for them to fill gaps in the paperwork, and have the entire cost paid by the government.
Is this right or wrong? I'm mixed on it. But it's policy, and every president has done it.
