No, electrons are not distributed evenly over a molecule. Electrons are arranged in shells around the nucleus of an atom, and the shells are filled in a specific order. The first shell can hold up to two electrons, the second shell can hold up to eight electrons, and so on. The electrons in the outermost shell are the most loosely held and are therefore the most reactive.
In a molecule, the electrons are shared between the atoms. However, the electrons are not shared equally. The atoms that are more electronegative (have a higher affinity for electrons) will attract the electrons more strongly. This can result in a polar covalent bond, where the electrons are not evenly distributed between the atoms.