In American English, we use the auxiliary verb should with all three persons (first, second, third) to express a sense of duty. Thus:
She should study more diligently.
We use would with all three persons (first, second, third) to express a regular practice, a hypothetical, a preference, or a future in a past statement. Thus:
Each day, the police would form a perimeter around the White House. (regular practice)
He would write a book if he could. (hypothetical)
We would like you to come to the party. (preference)
I said that I would complete the report later. (future in a past statement)
If your colleagues insist on using should with the first person when they mean “would,” they are simply trying to sound British. Thus, they might say:
We should like you to come to the party.