Make blog page private

We are unable to duplicate your issue.
Using WP 4.53 and BP 2.6 and a standard WP theme such as 2015.
Any Post or Page that is checked ‘private’ cannot be viewed by a logged-out user.

Perhaps there is a conflict with your theme?
Try switching momentarily to a WP theme like 2015.

Perhaps there is a conflict with some other plugin on your site.

I am using a standard theme 2012.

I use BP registration options could there be a conflict with that.

I have tested with other pages including the archives page and it works as it should but not for the blog page. I am using a static front page and a separate blog page with posts. Is there some thing special about that page?

I did some more testing and created a new page set it to private and tried to view while logged out. Everything worked as expected and I was redirected to home page. Then I set that page as my blog page in the customizer and it immediately became accessible while stil being logged out.

Ah, those pages are excluded.
Otherwise a redirect loop would be created.
We may be able to include the blog page and only exclude the front / home page.
We will try to address this in the next couple of days.
Thanks for the clarification.

Thanks for the reponse and taking time to look at possible solutions

We took a look at this.

Then I set that page as my blog page in the customizer and it immediately became accessible while stil being logged out.

By ‘blog page’ do you mean this setting?
Front page displays > Your latest posts

If so, then that will always be public regarding of the ‘private’ setting option.

Regardless of which option you select in the customizer:
Front page displays: Your latest posts OR A static page
That page will be public.

Otherwise a redirect loop would be created for 99% of users.

I have static front page I expect and want that static page to be public. I set a different page as the blog latest posts page. Are you saying the latest posts page will always be public despite setting that to private using the plugin?

For your situation, you could try hacking:
bp-simple-private\inc\pp-private-front.php

Change this:
if ( is_front_page() || is_home() || bp_is_register_page() || bp_is_activation_page() )

To this:
if ( is_front_page() || bp_is_register_page() || bp_is_activation_page() )

Or maybe this:
if ( is_home() || bp_is_register_page() || bp_is_activation_page() )