There doesn't appear to be an error in the log entries that you have, so it may just be that the logging levels between the two nodes are inconsistent (and the same thing is happening on each).
The Publisher Service logging is enabled in the Tridion MMC snap-in (under Publisher settings).
Once this is done, the location and logging level is done in the following location: [Tridion]\bin\TCMPublisher.exe.config
According to Elena's logging blog post, logging from the transport service is done in the [Tridion]\config\logback.xml file and output to the cd_transport.yyyy-mm-dd.log log file (by default).
<appender name="rollingTransportLog" class="ch.qos.logback.core.rolling.RollingFileAppender">
<rollingPolicy class="ch.qos.logback.core.rolling.TimeBasedRollingPolicy">
<fileNamePattern>${log.folder}/cd_transport.%d{yyyy-MM-dd}.log</fileNamePattern>
<maxHistory>${log.history}</maxHistory>
</rollingPolicy>
<encoder>
<pattern>${log.pattern}</pattern>
</encoder>
<prudent>true</prudent>
</appender>
UPDATE [24/06/2013]:
It appears that the question has been renamed to "Two publishers: one deletes remote files, the other one doesn't".
Are you definitely sure that this is happening? If an item is published, and added to the publish queue, then only one of your publishing nodes will pick this up and process it. (Leaving the other node waiting for other published items to process.)
Your log file entries do not seem to suggest that there is an error removing the remote files. Are your files still present on the filesystem of the machine running the Deployer? If so, then, depending on what files are being left, that it is the Deployer failing to clean them up. You may want to check that the Cleanup attribute is set to true in the following node on your cd_deployer_conf.xml file:
<Queue>
<Location Path="D:\Tridion\Incoming" WindowSize="20" Workers="20" Cleanup="true" Interval="30s"/>
</Queue>
Can you update the question with more details about exactly what the issues you are seeing are please?
