Also it isn't just the Secret Service, in the USA it is a federal CRIME to threaten the President:
THREATS AGAINST PRESIDENT - 18 USC 871, makes it a Federal crime or offense for anyone to willfully make a true threat to injure or kill the President of the United States.
A person can be found guilty of that offense only if all of the following facts are proved beyond a reasonable doubt:
First: That the person uttered words alleged to be the threat against the President;
Second: That the person understood and meant the words he used as a true threat; and
Third: That the person uttered the words knowingly and willfully.
A "threat" is a statement expressing an intention to kill or injure the President; and a "true threat" means a serious threat as distinguished from words used as mere political argument, idle or careless talk, or something said in a joking manner.
The essence of the offense is the knowing and willful making of a true threat. So, if it is proved beyond a reasonable doubt that the person knowingly made a true threat against the President, willfully intending that it be understood by others as a serious threat, then the offense is complete; it is not necessary to prove that the person actually intended to carry out the threat.
I guess the Secret Service investigates to see if it was a true threat, or not.