RAW It does not give the Rogue a spell casting DC outside of the spells directly related to the feat
You have a Spellcasting Ability, not a Spellcasting Class Feature
Magic Initiate states (PHB, Chapter 6-Feats) with my emphasis:
Your spellcasting ability for these spells depends on the class you chose: Charisma for bard, sorcerer, or warlock; Wisdom for cleric or druid; or Intelligence for wizard.
The spellcasting ability is directly related to just those spells and are a result of the class you chose when taking the Feat. However, this does not grant you the Spellcasting Class Feature.
But why not? After all they are a Magic Initiate!
Other than the strict reading I provided above, I'm not sure why it would be a big deal to allow the Rogue to have a spellcasting DC for those types of items that require it.
The feat is called Magic Initiate, after all. They have been initiated into Magic of a certain Class. Let the Rogue have a spellcasting ability in that class in general and utilize the associated ability score for magic items that require it.