When participant company is adding a Dataset they do not see the other partner as an option to share under the drop-down Share with relationship(s).
Share with relationship(s) -> Select one or more relationship(s)
Only the relationship name will be shown as an option.
The name can be confusing, as it may only show one name.
Logged in as Participant company (Paper Street Soap Co.) you can only see your company as an option:
Check the name of the relationship first, this is what you will be sharing with.
Otherwise change the name of the relationship, to make it easier to find.
You can edit the relationship name to have both the Lead and the participating company.
Now the relationship has both partner names and is easier to find:
Note: The relationship name is shared between you and your partner. If/when you make changes to the relationship name, make sure you coordinate any name changes with the Lead company.