Yes, it needs to be signed by the people making the agreement or representatives authorized to make the agreement on a company's behalf. You are authorized to write any document that can be recognized as valid and enforceable in a court of law as long as it follows any statutes and is valid and legal.

Can I write by hand on stamp paper?

Whether the same is written in hand or typed does not matter. As long as the terms are clear, and understandable it should not be a problem. Please ensure in case there are any changes or alterations then the same are counter signed by both the parties.

Is anything written on stamp paper valid?

They are valid as per current law, but future is doubtful:

Although the use of old stamp paper is valid even after six months of purchasing it, still it is better that you must either deposit it back to collector within six months and get your refund from him or use it as early as possible.