from your description it seems that your sendmail server can send mails to that server which doesn't need authentication.
The one which which denies to give service requires authentication.
I suggest following methods to solve this problem.
edit your /etc/mail/sendmail.mc file aas following.
add following line
FEATURE(`authinfo',`hash -o /etc/mail/authinfo.db')dnl
after this line.
save changed and exit.
cd to /etc/mail
create a file as
AuthInfo:10.2.56.47 "U:user" "P
( change IP address of your SMTP server, change user with real user on that SMTP server and change password with real password of that user.)
save and exit.
now execute following commands.
#makemap hash authinfo < authinfo
#m4 sendmail.mc > sendmail.cf
now restart the sendmail service.
#service sendmail restart
Now try sending mails through this SMTP server. It should work.
Please let me know the result.