Are you sure you're referring to the profile newsfeed and not dashboard one? Because we have just tested it and couldn't reproduce that. 

Can you please provide the screenshot of the page, where you see the forum post from private group in the member profile newsfeed?

I am definitely referring to the profile newsfeed, not the dashboard. A user who is not a member of a private group provided this screenshot that he took from the user stevita's profile. I can confirm I see the same thing with my test account.

HW, what is your software version?

I would really appreciate if you try to reproduce the same issue on our demo website. Because we cannot. Thanks Keep us updated 

I am using 1.8.0. I am on vacation just with a mobile device, but I did try the demo site. A private group post I made didn't even appear to myself on my own profile.

I am totally baffled why it is different on my site or what setting I could have changed to make that happen.

HW, do you use any 3rd party plugins, please disable them all switch to the default theme and see if the issue persists. If it does, please PM me your admin website access details, Cpanel access details, I'll take a look at the issue. 

My apologies for the delay in getting back to this.  I returned from my vacation to find my apartment had flooded, so my website took the backburner while I was dealing with that.

I turned off all plugins except the following:  blogs, events, friends, forum, groups, messages, newsfeed, photo, privacy, video.  I changed to the default theme.  The issue persisted.

Shall I send a message?

Yes, please

I'm sorry but I can't reproduce the issue. See screenshot of the private group and profile page newsfeed. As you can see there's no any items in the profile newsfeed regarding private group. 

As I said you probably referring to the dashboard newsfeed where you can see these activities. However dashboard is accessible only by the owner of the account. See 3rd screenshot

First, thank you for looking into this.

Second, I am definitely not confused about the difference between the dashboard and profile newsfeeds.  I know I am not.  However, after your test, I know that this is only happening with REPLIES to forum topics in private groups, for whatever reason.  I started a new topic and did not see it appear, yet the reply did.  Please see my attached screenshots.  The first confirms the group should be private, and the second shows the behavior I have been observing with the two private groups on my site.

EDIT:  Adding third screenshot so you can see it was taken from a regular free/default account that should not have access to the private group . . . yet can view the post on my profile page.

ok, we were able to reproduce the issue, we have passed it to our devs. thanks for the report