When executing a share transfer agreement between the seller and the purchaser, does it need to be witnessed or notarized?

It does not need to be either. Generally speaking, contracts do not need to be witnessed or notarized.

I agree with Mr. Goodman. Contracts generally are not witnessed or notarized, unless they fall into a few special categories: wills, real estate transaction, public company stock certificates, etc. Unless you are worried that the seller may later claim that an impostor signed away his company, I would not bother with notarization or witnessing.

Agreement does not but the share certificates might.