[exim] Could i have two servers with the same domain?

Hello:
I want to know, if i have two servers with the same domain
(domain.com), but just one is up (ip address is targeting at the
specific computer) i will get trouble sending and receiving email,
because on both servers, it will find "local_domain" (same domain) but
it´s not local because the computer up is the other one, so it will
think the domain is local. The other one is working as backup system.
Please, if you know something about it, send me an email,
Thanks in advance,
Roberto