This is not possible in general because the files in your project A are probably on a different virtual machine than your project B. However, there is the rare possibility that they happen to be on the same machine. In this case, because SMC is just Linux, the following may be possible after you open a terminal in both project A and B:
projectA> ls -l $HOME/filename
-rw-rw---- 1 xyz xyz 31 Jun 21 11:59 /projects/xyz/filename
projectA> chmod o+r $HOME/filename
projectA> ls -l $HOME/filename
-rw-rw-r-- 1 xyz xyz 31 Jun 21 11:59 /projects/xyz/filename
... where xyz is the project ID of project A. Now, in that rare case you should be able to do from the project B terminal:
projectB> cp /projects/xyz/filename .
Don't forget to change back permissions with:
projectA> chmod o-r $HOME/filename
projectA> ls -l $HOME/filename
-rw-rw---- 1 xyz xyz 31 Jun 21 11:59 /projects/xyz/filename
See also the Wikipedia page about chmod.
I assume this is a Sage math cloud question?
I've been just downloading and the uploading into the second project. I think there is a more efficient approach using github. But, as I understand it at the moment, you cannot copy between projects in SMC.