You won’t be “on a different local network,” you’ll be accessing specific networks (or subnets) via the VPN tunnel rather than some other network interface on your machine.

So if you’re at home with a 192.168.0.0/24 network and you want to access an office resource on the 192.168.141.0/24 network, likely what will happen is your machine with have a route to 192.168.131.0/24 via the network the VPN provides (let’s just say 10.0.0.1).

Depending on how everything’s configured, the server you’re accessing might see it coming from the VPN server (masquerade) or it could very well be passed on as-is (which would only work if the server has a routing table back to you via the VPN).

Typically when people use VPNs for internet access, the traffic is sent out masqueraded so that it appears to come from the VPN’s WAN IP address.