The left.
This has been the standard for hundreds of years. Recently people have mistakenly worn them on the right (which is for name tags, so that when people shake your hand they can read your name, or have a visual reminder as introductions are made). You will notice that the Queen of England always wears her pins and brooches on her left side.
Additionally, they need to be placed appropriately, which is approximately half way between the shoulder and bust-line for women (or balanced if the garment limits the ability to do so). Shapes should be placed so they are visually pointing up, animals should face the left shoulder, and birds, etc., should be turned so they aren't "flying into the face," etc. Small lapel pins (denoting affiliations) should be centered upon a lapel. Pins that anchor scarves can be center front or again, pinned on the left side.
On the left side, always
It should be worn over his heart.
He wears a US flag lapel pin. Please note that lapel pins do not make one more or less patriotic.
left side lapel above notch
Order of Australia lapel pin. From wiki- “a gold lapel pin for daily wear is issued with each badge of the order at the time of investiture; AK/ADand AC lapel pins feature a citrine central jewel, AO and AM lapel pins have a blue enamelled centre and OAM lapel pins are plain.”
left
It looks like the pin for Lions Clubs International
across the chest or across the butt
This is more of a fashion-oriented question. When it comes to politicians, I would say no. Politicians are to wear one lapel pin: the American Flag.
AOPA Lapel pin
The lapel pin goes in the button hole that is sewn in the lapel. If there isn't one, then one goes about one inch down form the notch in the lapel and attaches the pin in that area. In other words, the pin should be attached about one inch down from the tie knot and about 4 inches over on the lapel. This pin should be a statement of some type, a religious symbol (if your position allows it) or a cause. It should be small, unobtrusive, and tasteful.
where to buy great lakes lapel pin