I ran the example below in MySQL running on UNIX. First I created 2 users with the same password. Then I looked up their encrypted passwords and they were both the same. This suggests to me that the encrypted value comes from the password alone: 

UNIX > mysql -u root -p
Enter password:
Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 8846
Server version: 5.0.67 Source distribution
 
Type 'help;' or '\h' for help. Type '\c' to clear the buffer.
 
mysql> create user 'joe'@'localhost'
    -> identified by 'bloggs';
Query OK, 0 rows affected (0.01 sec)
 
mysql> create user 'fred'@'localhost'
    -> identified by 'bloggs';
Query OK, 0 rows affected (0.00 sec)
 
mysql> select user, host, password
    -> from mysql.user
    -> where user = 'joe'
    -> or user = 'fred';
+------+-----------+-------------------------------------------+
| user | host      | password                                  |
+------+-----------+-------------------------------------------+
| joe  | localhost | *C928C1952D09F99240B3B782A37E92E6C5E13913 |
| fred | localhost | *C928C1952D09F99240B3B782A37E92E6C5E13913 |
+------+-----------+-------------------------------------------+
2 rows in set (0.00 sec)
 
mysql>