He is supposed to be here by now. He agreed the precise time with me yesterday.

He is supposed to be here by now. He agreed the precise time with me yesterday.

First, I also feel that "he agreed the time with me" sounds very strange to American ears.  "He agreed on the time with me," or "we agreed on the time" sounds better.  (I have no problem with "we picked the time" or "we chose the time" but "agree" does not seem to want to take a direct object in AmE.  You can agree on something, or agree to do something, but not just agree something.)  However, I'll certainly take your word for it that it sounds fine in BrE. 

Going back to the original question - in the two examples above I would not see anything wrong, or any change in meaning, if you switched "should be" and "is supposed to be" in the two sentences.  Eimai Anglos, could you please explain what you see to be the difference?

I'm trying to figure out if I use "he should" and "he's supposed to" differently according to any pattern that I can explain.  Here's what I think, although  it's just intuition, not any rule I ever learned:

To express a moral  or ethical imperative, I would be more likely to use "should."  We should be thoughtful of the feelings of others.  You should take better care of yourself.

To express expectations imposed by another person, I would be more likely to use "is supposed to."  You're supposed to clear the table before you go out to play.

To indicate "one would reasonably expect that..." I would use either one.  It's nine o'clock - he should be here by now.  It's nine o'clock - he's supposed to be here by now.

I'd like to hear other opinions as well.

Thanks, davkett!  I have one more to add, that my daughter reminded me of - occasionally, "should" is used to mean "probably, most likely" without any implication of obligation or requirement.  This "speculative" sense of should is very hard to distinguish from the "obligation" sense without intonation and context.  For instance, I might say to my husband, "where are your car keys?" and he might reply "they should be on my desk somewhere," just meaning "they are probably on my desk."  Then, after I have borrowed them and forgotten to replace them, he might say "where are my car keys?  They should be on my desk, you know!" meaning "they ought to be on my desk where they belong!"  In this case, where "should be" is used to mean "very likely is" you cannot substitute "supposed to be."

My daughter is my best consultant for tricky questions like this.  How many 16-year-olds do you know who are willing to discuss grammar and semantics with their mothers during an evening stroll?  I consider myself very fortunate that she share my interest in this sort of thing.