What particle may be shared by or transferred to other atoms?
Electrons are transferred to other atoms in the formation of ionic bonds, or shared by atoms to create covalent bonds.
Of the three main components of atoms, protons and neutrons are bound (by the strong nuclear force) tightly in the nucleus, and generally play no role in determining the chemical nature of materials (aside from determining which element or isotope we are considering, of course).
On the other hand, the interaction between the nucleus and the electrons is governed by the electric force (electrons do not experience the strong force) and especially for the valence electrons of an atom, this attractive electric force can be weak enough that application of ordinary amounts of energy can cause changes to the valence shell of an atom when two or more atoms interact.
These changes generally involve transfer of electrons (from a metal atom to a non-metal) or sharing of electrons (typically between two non-metals).
The transfer of electrons results in what is known as an ionic bond between atoms, while sharing of electrons creates a covalent bond.