should, would

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.