Inside the chimney, there is a smoke shelf collecting rain water and let it flows to the backend of the chimney.
Next to the Smoke shelf, there is a damper which can be opened or closed from inside fireplace.
The smoke shelf may collect some debris.
You may access to the chimney damper through a fireplace.
The pictures below show the bottom up view from the fireplace.
In Winter time if you don't use the fireplace, you may close the damper to prevent cold air or rain water coming in.
Chimneys should have caps like the pictures below.
Or at least the Chimneys should have some protection using the materials shown below to prevent debris or birds coming into the chimney.
You may need to get on the roof to inspect the chimney.
Inspect the following areas to see if there are any cracks and leaks:
Flashing
Motor joints
The Crown
See the pictures below as reference.
Troubleshoot Chimney Leaks through fireplace: https://sites.google.com/view/gordianpropertiesllc/gpl-repairs/Troubleshooting/Chimney-Leaks