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1. A document signed by the two parties (lessor and lessee -- owner and renter) is valid (official/formal in your words). Each party should have an original. That means that each of you should sign both since photocopies of signatures are not valid.
2. Notary/notarization: the person who signed the document shows proof to the notary that s/he is the person whose name appears on the document and who states to the notary that s/he is the person who signed. That's a 2-parter: you show you license to the notary and say, I signed the document. The notary doesn't have to witness the person actually signing it. Some single signature documents should be notarized - such as a POA - and that's why a contract with 2 signatures doesn't need to be notarized. Each signatory to the contract "proofs" the other in that the contract is not valid without both signatures.
You need to have an attorney who represents you look over these documents - although it is too late - you've already agreed to the terms. But you do need someone to adequately explain what you agreed to.