Runi journals function just like any other breedable journal! They are the perfect spot to display your rsuchu (as is technically required in the shop rules) and it's familiar, along with any items it may have collected.
You are welcome to post any information about your rsuchu and it's adventures. Rsuchu with themed or narrated journals might have a better chance of throwing offspring with rare traits! Roleplaying and diary entries are absolutely not required, but you are welcome to keep them if you so desire.
Owners are encouraged to keep one journal thread with all of their runi in it, unless you plan on making a really epic journal for an individual runi; then by all means, give it it's own thread!