I did not setup D system so I am not sure how.
What do you mean ? Your replies are confusing.
If I try to access /usr/sap share on P system from D system as user <D-SID>adm I am prompted for password. How is possible then that D system can write to /usr/sap/trans on P?
If you want to configure a 3 system landscape and use a common transport directory for these systems then you need to make sure that the transport directory is accessible for all the 3 systems by the sidadm and SAPServiceSID users.
Best practices for configuring transport directory - Basis Corner - SCN Wiki
Specifying the Central Transport Host (SAP Library - Transport Tools (BC-CTS-TLS))
Regards
RB