On a Linux system, for example, if you're allowed to look inside the process, which would be for similar reasons to why you're allowed to look inside a configuration file belonging to the user, then it's pretty much just as easy.

Suppose we want to look inside process #1234, we just open the file /proc/1234/mem and suppose we know we want the value at address #003e0f20 we just seek to that address and read however much we want.

Unlike a typical data file the memory map has holes in it, so we do need to know where we're going, but that's OK the Linux kernel also provides information about what is in there and where it is, so e.g. we can root around specifically in the program's heap.