What's the best way to check for permissions at runtime using MVP architecture?

What I would do is:

The view will implement:

public Activity getViewActivity();

The presenter will implement:

public void requestPermissions();
public void onPermissionsResult();

Inside requestPermissions, the presenter will do: getViewActivity().checkSelfPermission; getViewActivity.requestPermissions(); etc.

The view will call inside the onRequestPermissionsResult callback to presenter.onPermissionsResult();

With this all the logic will be implemented inside the presenter.

In my opinion, your presenter is decoupled: it won't depend on any view implementation (it will only depend on the view interface).

"I've heard that keeping your presenter free from Android code is good for testing." I don't understand this part. If the code is good, it can be tested without any problem.

If you still want to be able to mock permission access/requests, you still create something like a PermissionHandler, but only reference it inside your view class. For example -

Interface:

public interface PermissionsHandler {
    boolean checkHasPermission(AppCompatActivity activity, String permission);
    void requestPermission(AppCompatActivity activity, String[] permissions, int requestCode);
}

Production implementation:

public class PermissionsHandlerAndroid implements PermissionsHandler {
    @Override
    public boolean checkHasPermission(AppCompatActivity activity, String permission) {
        return ContextCompat.checkSelfPermission(activity, permission) == PackageManager.PERMISSION_GRANTED;
    }

    @Override
    public void requestPermission(AppCompatActivity activity, String[] permissions, int requestCode){
        ActivityCompat.requestPermissions(activity, permissions, requestCode);
    }
}

Mocked class (for example, to test and make sure your activity correctly handles onRequestPermissionsResult)

public class PermissionsHandlerMocked implements PermissionsHandler {
    @Override
    public boolean checkHasPermission(AppCompatActivity activity, String permission) {
        return false;
    }

    @Override
    public void requestPermission(AppCompatActivity activity, String[] permissions, int requestCode){
        int[] grantResults = new int[permissions.length];
        for (int i = 0; i < permissions.length; i++) {
            grantResults[i] = PackageManager.PERMISSION_GRANTED
        }
        activity.onRequestPermissionResult(requestCode, permissions, grantResults);
    }
}

Then in your activity:

PermissionsHandler permissionsHandler;

@Override
protected void onCreate(@Nullable Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);

    permissionsHandler = Injection.providePermissionsHandler();
    //or however you choose to inject your production vs mocked handler. 
}

//method from your view interface, to be called by your presenter
@Override
void requestLocationPermission() {
    permissionsHandler.requestPermision((AppCompatActivity) this, new String[]{Manifest.permission.ACCESS_COARSE_LOCATION}, REQUEST_CODE_LOCATION};
}

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fobo66, you can always make the view implement much more generic methods like checkLocationPermissionGranted() and requestLocationPermission(). Then your view implementation can reference the activity as needed, and your presenter never has to touch the activity reference.