What specific activities do you do as PM to develop skills of your team members?

Most functional managers I've worked with were grateful that their employees were getting extra attention and care. On rare occasions I've had a functional manager come to me and say, "I don't want him/her doing that," to which I would say, "Oh. Okay. no problem. I'll take care of it." Then I would tell the employee, apologize for the lost opportunity, sympathize for a bit, and we would both go on with our lives. This approach has historically kept the team member engaged and functioning well as part of the project team, and dispelled any potential friction with the functional manager.
As to your second question, Adrian, I have managed people whose skill sets I do not have, but whom did not have some of my skill sets, either. Whether it's soft skills, time management, emotional management, public speaking or whatever, there's much to share about being a more well-rounded person, employee and leader.

Take the time to understand what the professional hopes and dreams are for your team members and try to find opportunities to connect the activities they will perform on your project to those aspirations.

I've sometimes had team members whose people managers ignore their personal development who have appreciated "someone" actually asking them what they would like to do!

Kiron

The issue Lenka is the temporary nature of projects, and the scope of responsibility outside the deliverables of the project which may require things like training etc. Otherwise invariably such skills development falls into the hands of functional departments.

So for once, I might have to agree with Adrian, kind of hehe

Thank you Adrian, I see what you mean and that’s all very valid points. There must be a balance apparently and PM should be aligned with functional managers if there is considered change or adding to specialization to expert professions.
But apart from that there are also opportunities for team members to learn new things that are coming from the project work, such as team work, conflict resolution, time management.

I agree, if the skills that the PM wants to develop in the project team members are in addition to the professional (hard) skills then yes this could work.

For instance some technical experts may be very good at their work but lousy at communicating or working in a team, for certain PMs may help if they have the capacity to teach others these soft skills.

Thank you for this message now I understand want you wanted to say and I agree. As long as the PM does not interfere with the professional skills development, which is the the duty of the functional manager, then most likely there will be no conflict.

PMs could also work on the development of the professional skills but only if they are themselves very experienced professionals in the line of work of the team members. Otherwise they should stay away from this and eventually just focus on soft skills development.

Challenge employees to go beyond their skill sets and try new things. Allow them to make mistakes. Let them explore areas that pose manageable risk to the project. Be open to their wishes to explore. If an employee says, "I could take a crack at it," let them try. Team them up with more skilled team members who can guide them and have their backs. And trust them. People want to make a meaningful contribution. Give them the space, and they will.