Both of us won a prize

Both is plural ( Both of us are married ) 

Both Argentina and West Germany have won the World Cup twice.

(Argentina has won the World Cup twice and West Germany has won it twice.)

As for the sentence you mentioned

Snappy"Both of us won a prize"

It has on

YoungBuddyBoth is plural ( Both of us are married )

Both Argentina and West Germany have won the World Cup twice.

(Argentina has won the World Cup twice and West Germany has won it twice.)

As for the sentence you mentioned

Snappy"Both of us

Each of us has a desk.

Both of us have desks. [This could be confusing in that each of us has more than one desk, but context would probably take care of it.]

Each is always singular 
For example 
Each of the students is responsible for doing his or her work in the library.

( Each is singular )  that's why  ( Each is )
Don't let the word " students " confuse you, the subject is " each ".

It's the same for your sentence 

Snappy"Both of us won a prize"
Does the above sentence have a double meaning?

I would not say so. I don't take "both of us" to be equivalent to "we". I take this sentence to mean only "You won a prize, and I won a prize" or "He (or she) won a prize, and I won a prize".

The "you", "he", or "she" part is ambiguous, but I don't get the impre

CalifJim

Snappy"Both of us won a prize"
Does the above sentence have a double meaning?

I would not say so. I don't take "both of us" to be equivalent to "we". I take this sentence to mean only "You won a prize, and I won a prize" or "He (or she) won a prize, and I won a prize".

Thanks. If I change the sentence s

YoungBuddyIt has only one meaning, that each one of us won a prize.

I see this differently. "Both of us won a prize" means you both won the same, single prize. "Both of us won prizes" means each one of you won a different prize.

This same topic has been discussed a few days back, in the context of a group raising their hand or hands.

SnappyThanks. If I change the sentence structure to "We both have a desk," it still means "Each of us has a desk." Is my understanding correct?

I'll support CJ here, so yes. And if you say «Both of us have cats», it will mean that each of you have more than one cat.

Anton

ferdisI see this differently. "Both of us won a prize" means you both won the same, single prize. "Both of us won prizes" means each one of you won a different prize.

I agree to some extent, because I believe that " no sentence is an island " ... so we should always take the context into our consideration, If not so, we will be interpreting in many various way

Ant_222And if you say «Both of us have cats», it will mean that each of you have has more than one cat.

Exactly. That's why a man, when referring to his wife and himself jointly, says "We are married", and never "Both of us are married". "Both of us are married" includes the case where each of the two are married, but not to each other.