How to solve it, the contents of private messages Iraj not included in this notification system that sends e-mail? The problem with this is that members reply to a private message but if you do it via e-mail, you will not receive a private message sender, but the admin. This is a serious problem. How can this be avoided?

Hi,

every notifications, including private message. that "only" notification. you cannot reply it via email, but must from jomsocial message interface.

I suggest you to put some explanation about that at email notification content by override this language constant:
COM_COMMUNITY_EMAIL_INBOX_SENT

from joomla backend > extension > language manager > overrides

maybe you can put text "please click link below for replying"

Regards

Thank you, but this is not the best solution for me. I would like to not appear in the e-mail message content. Thus forcing the user to log in to the site, if you want to read the message.

Hi Kertész,

you can do it from from language overrides.
goto joomla backend > extension > language manager > overrides
select language section at Filter > create new

search this COM_COMMUNITY_EMAIL_INBOX_SENT constant

click on search result, and you can changing the text for email message content

changing it from

Hi {target},\n\n{actor} just sent you a new message.\n\nTitle:\n%1$s\n\n%2$s\n\n\n\nYou can also read your new message on the site here:\n\n<a href="{url}">{url}</a>\n\n\nHave a nice day!

Hi {target},\n\n{actor} just sent you a new message.\n\nTitle:\n%1$s\n\nYou can read your new message on the site here:\n\n<a href="{url}">{url}</a>\n\n\nHave a nice day!

Thank you very much!
This was what I needed. It works perfectly.