If gmail is used with a local client (like Thunderbird, Outlook, etc.) then the email header should have the private IP address. Note that it is possible that some of the information is stripped by the client or client network before reaching the SMTP server. Take a look below:
----- Mail sent from Thunderbird using googlemail SMTP -----
Received: from [10.0.0.101] ([188.8.131.52]) <--- here you can see the private (10.0.0.101) and public (184.108.40.206) IP address of the sender connecting to the SMTP server.
by smtp.googlemail.com with ESMTPSA id <-- this line tells you that the message was received by SMTP
for <[email protected]>
Mon, 02 Nov 2015 23:01:38 -0800 (PST)
To: Joshua James <[email protected]>
From: "Joshua I. James" <[email protected]>
If the email is sent from the Gmail web interface (in the browser), the private IP address is NOT available. Google's server only sees the suspect's public IP address access the google web server.
------- Sent from gmail web interface ------
Received: by 10.50.10.233 with HTTP; <---- "with HTTP" means received via web interface on server 10.50.10.233 (google). The sender's IP is not shown.
Date: Tue, 3 Nov 2015 16:08:03 +0900
From: "Joshua I. James" <tes[email protected]>
To: "Joshua I. James" <[email protected]>
If the header is only showing google's address, then the suspect must have been accessing the web interface (check for "with HTTP"). In that case, google will only have the public IP of the suspect.