[OpenIntro]

When these two react, I know that a hybrid compound is formed, with a positive charge on the chlorine that is bonded to the now negatively charged boron. My question is, who attacks who? Is it chlorine attacking the empty side of Boron and sharing the electrons?

[movies]

Do you mean the reaction with Cl₂ or with Cl^- ions?  In either case, I don't think that there is a positive charge on the chlorine attached to the boron.

With respect to "who attacks who" by convention electrons always do the attacking, so it would be the electrons from chlorine (or chloride) that attack the vacant p orbital on boron to form the tetrahedral complex.  The attack can come from either side because BF₃ is trigonal planar.  The product is probably a covalently bound complex, so sharing of electrons.  The electronegativity difference between B and Cl is boderline for ionic vs. covalent, but the B is already electron deficient from the 3 Fs.

[Donaldson Tan]

The B-Cl bond formed would experience charge seperation, in virtue of being a dative bond

Chlorine contains the required lone pair for the formation of B-Cl bond. It acts as the nucleophile. Boron in BF₃ is electron deficient, so it acts an electrophile.