Learn Hardcore Russian: На каждом договоре стоят подписи и печати сторон. - On each contract, the parties’ signatures and stamps are present.

на каждом договоре literally means “on each contract (document)”, focusing on the physical surface or page where signatures and seals are placed.
в каждом договоре would mean “in each contract (text)”, implying something inside the content or clauses of the agreement.

Since we speak about where the signatures and seals literally stand (on the paper), на is correct.

What case is каждом договоре and why does it end in -е?

каждом договоре is the singular prepositional case of a masculine noun with its modifier:

каждом is the adjective каждый in prepositional singular masculine (ending -ом).
договоре is договор in prepositional singular (ending -е).

The prepositional case is required by the preposition на when it indicates location rather than direction.

Why does the sentence use the verb стоят (“stand”) instead of есть (“there are”) or находятся (“are located”)?

In Russian you often use стоять or лежать with printed or stamped elements to indicate they are affixed or appear on a surface. Saying подписи стоят or печати стоят emphasizes that the signatures and seals are physically “standing” on the document. Using есть (“there are”) would be grammatically possible but less idiomatic, and находятся would sound overly formal or impersonal.

What case and form is сторон, and why is it needed?

сторон is the genitive plural of сторона (“party” in a legal sense). We use genitive here to show possession or association: подписи сторон = “signatures of the parties.” Since there are at least two parties to a contract, the plural genitive сторон is used.

What does печати mean here, and why is it plural?

печати is the nominative plural of печать, meaning “stamp” or “seal.” In many legal systems each party has its own seal, so you end up with multiple seals on the contract. Hence plural.

How can I tell whether печати is nominative plural or genitive singular (they look the same)?

Context and syntax:

As subjects of the verb стоят, подписи и печати must be in the nominative case.
You can’t have a genitive singular as the subject of a finite verb without an explicit impersonal construction (e.g., нет печати “there is no seal”).

Where is the stress in договоре?

The stress falls on the third syllable: до-го-ВО-ре (prepositional form).

What is the subject of the sentence, and how does the verb agree with it?

The compound subject is подписи и печати (signatures and seals). It’s plural, so the verb stands in 3rd person plural as стоят.