The only safe way of doing this is to mount the filesystem as read-only on both hosts.
The buffer cache is a memory-based data structure that sits between the data on the disk and the apllication. When an application makes an io request, it first checks to see if the data are already in the buffer cache and if not then the data is read from disk. Similarly, when an application does a write, it writes to the buffer cache and sometime later this write is actually done to the disk. The buffer cache makes the i/o operations more efficient and faster.
Now suppose that system A does a write to the filesystem. It does the write and that information is stored in the buffer cache but system B has no way of knowing that and reads from the disk. If you attempt to mount the filesystems read-write, the best result you can hope for is occasional stale/invalid data; the much more likely result is corrupt filesystems on both hosts.
One option you have is to use the LVOL as a raw device but even there there has to be a synchronization mechanism to prevent both hosts from writing to the same block simultaneously.
If you are using filesystems, again, the only safe method is read-only on all the hosts.
If it ain't broke, I can fix that.
