Yes and no. In fact there are two ways of accomplishing this.

I'll start with the no part. There is no way to have both a private and a public set of conversations simultaneously in a club. The reason for this is that club permissions are managed by the type of club that the club owner has opted to create. There are five of these types of clubs, as follows.

Public 
Everyone can see the club and its posts, and can participate without joining.

Open 
Everyone can see the club and its posts, but only members can participate. Everyone can join.

Reserved 
Everyone can see the club and who's in it, but only members can see posts and participate. Users need to ask a leader to join.

Private 
Only members can find the club and see its posts. Users need to be invited by a leader to join.

Read Only 
Everyone can see the club and its posts, but only members can participate. Users need to be invited by a leader to join.

Most clubs opt for the Reserved type which means that people can see that there is a club, they can see the members, but they cannot actively see or participate in content without being members. If one chose, one could create an Open club in which case non-members could see all the content but not participate. And then there is the more drastic option, one that we do not allow for chartered clubs, and that is one in which everything is open. As you can see, there are two other types, Private and Read Only.

Now for some reasoning. The idea behind Ham Community is to encourage both private conversations among club members, but also the opportunity to converse with the broader amateur radio community. As such, a club member who wishes to start or engage in a more open conversation, would simply post the topic in the general community. This paradigm encourages closed discussions within the club and open discussions with the community at large.