In my experience the answer is, "It depends..." It depends on what VPN
software you are using. And it depends on how your VPN software is
configured.
For example, my employer uses a Cisco VPN. My employer has it configured
such that when I have the VPN connection active I cannot make a network
connection except through their internet connection (not through mine).
That means, for example,.that my NAS and other computers connected to the
LAN side of my router here at home are inaccessible to me when the VPN
connection is active -- but I can access most public internet sites.
My employer *could* configure the software to allow me to access local
resources when the connection is active but they have/will not as this would
open their network to attacks if one of my other machines has been
compromised. Because the VPN client inserts itself into the network stack
on my machine, it is able to wrap itself around all traffic and only allow
what it likes. So unless a packet is coming from the server side of the VPN
connection, it simply falls on the floor. And if I attempt to access
192.168.0.* (i.e., a local resource on my LAN), this gets wrapped up by the
VPN and resolves to nothing.
So if you want access to web sites and/or other resources on the internet
(but not on your local LAN) then it is generally possible to have both (all
your connections "typically" end up passing through the
firewalls/routers/security appliances on the other side of the connection).
If you want to have a VPN connection active and access resources on your
LAN, you will usually not succeed.
P.J.