The 12th Amendment states
The Electors shall meet in their respective states, and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves
Does this mean that if a President and a Vice President were to come from the same state, they could not constitutionally receive any votes from their home state?
This part of the 12th Amendment is a holdover from the prior rules:
The Electors shall meet in their respective States, and vote by Ballot for two persons, of whom one at least shall not lie an Inhabitant of the same State with themselves.
The Founding Fathers thought that if electors only had one vote, they'd each vote for a “favorite son” from their own state. The second vote would ensure that the President would have nationwide appeal.
In practice, we quickly ended up with national political parties instead of a patchwork of regional parties, so the requirement wasn't really necessary. But it was never changed. What did change was that Electoral College ballots now distinguish between President and VP votes.
What the Constitution says is that an elector of State X is not allowed to vote for both a presidential candidate who resides in State X and a VP candidate from State X.
But it's OK for a State X elector to vote for a State X resident for president and a State Y resident for VP. (In fact, most presidential candidates do win their home states.) Or vice versa: A State X elector can vote for a State Y presidential candidate and a State X VP. Just not both from State X. On the other hand, electors from State Y and State Z are perfectly free to vote for two State X candidates.
This is from the Constitution's viewpoint that assumes electors make their own decisions on whom to vote for. With political parties and pledged electors, it doesn't quite work that way.
So, what would happen if a party nominated a presidential candidate and a VP candidate from the same state? Without loss of generality, I'll assume that the state is Texas (and thus, that the party is the Republican party).
For the electors on the other side of the Sabine or Red Rivers, there would be no problem. None of them would be voting for a resident of their own states, so they'd meet the Constitution's requirement.
In Texas, however, electors would be prohibited from voting for both halves of their party's ticket. What they'd probably do is to vote for their party's presidential candidate, and become “faithless electors” when it came to the VP, either abstaining from that vote, or voting for a non-Texan (presumably, a different member of their own party) for that office.
Suppose that the election was close enough that Texas would swing it. The Republican presidential candidate would have an absolute majority of the electoral votes, and thus win the election, but no VP candidate would have a majority. The VP would then be elected by the Senate.
If the (newly-elected) Senate were controlled by Democrats, we'd end up with a President and Vice-President from different parties, and fights over the Secret Service budget.
To avoid this scenario (and for other “balancing the ticket” reasons), presidential candidates never nominate VP candidates from their own states.
The 12th Amendment is saying that the President and Vice President need to come from different states - not that the electors cannot vote for the "home state" candidate. Indeed, on average, most candidates receive their home state votes, and for much of the 19th century, this regional advantage was often considered when choosing candidates.
In fact, this was actually an issue in the election of 2000. While Dick Cheney had been a member of Congress from Wyoming, he was, in the year 2000, arguably a resident of Texas since he had property there. George W. Bush, governor of the same state, was as well.
In order to avoid the prohibition in the 12th amendment, Cheney had to quickly change his residency back to Wyoming (and sell his property), in order to avoid a lawsuit that had been filed. Eventually, a three-judge panel ruled that Cheney was, in fact, resident in Wyoming.
Had they not done so, Cheney would have been disqualified, and Bush would have had to name another person as his VP. (In actuality, he would have had to convince the electoral college to vote for another VP candidate, but I suspect that would have been formality, seeing as they were to pledged to vote for Bush.)
It says that they can’t both get votes from the same home state electors simultaneously. One of the two would need to give up votes should ‘they’ win.
