Something is charged if there is an imbalance between the number of protons and electrons. If it’s an atom or molecule, it’s called an ion.
If there are more protons than electrons, the item is positively charged. If it’s an atom or molecule, it’s a cation.
If there are less protons than electrons, the item is negatively charged. If it’s an atom or molecule, it’s an anion. If you’re familiar with word roots this is mnemotechnic, as “an” means “negative.”
What electrons would they share? There’s no extras! Depending on the specific “thing,” the charge may be spread out among several atoms (from “two” in a specific arrangement in organic chemistry, to “every atom in the chunk” in a chunk of metal); this applies whether the charge is positive (missing electrons/extra protons) or negative (extra electrons/missing protons).
One of the most common mechanisms in organic chemistry is having either a cation attach itself to a high-electron-density part of another molecule, or an anion attach itself to a low-electron-density part of a different molecule. That’s similar to what you’re thinking of, but in a much smaller scale.
One of the most common types of chemical bonds is the ionic bond, where ions with positive and negative charges stay close to each other so the global charge is zero, but each ion stays charged and doesn’t either hand over the extra electrons or grab some"body"-else’s. This kind of bond happens for example in table salt, NaCl, where the Na is Na[sup]+[/sup] and the Cl is Cl[sup]-[/sup]. That’s a molecule with a very simple formula; bicarbonate, NaHCO[sub]3[/sub] is more complex and its two ions are Na[sup]+[/sup] and HCO[sub]3[/sub][sup]-[/sup].