Jason kept a number of safe houses and small apartments open. Talia helped him get the system started with a variety of fake names and accounts that were nearly impossible to trace back to either of them. If any of the places were compromised, he could easily leave and know that it'd be taken care of. When he noticed Laura following him (again) after all of the fighting at Blackgate, he almost told her to get lost. But she wasn't from here, and she was caked in blood and dirt and possible injured. It was against his better judgement, but he offered her a place for the night where she could at least shower. If anything went wrong, he'd just use another safe house, knowing she wouldn't be able to find him.
There was also the issue of identities. She was open about giving a name, if it was a real one. Jason wasn't. He was even leery about removing his mask. But no one had seen him since... well, it had been a long time. Laura shouldn't know just by a face, and the name Jason Todd meant only so much to just a stranger. He was almost mad at himself for this predicament, but he couldn't just leave her. And she wasn't going to follow that kid home.
"Shower's that way," he pointed to Laura when they got inside. He propped his feet up on a table and immediately opened his computer to jot down notes and keep himself updated on Gotham. Not quite as fancy as the Bat's, but it got the job done. Reluctantly he took off his hood, still wearing a domino mask underneath. He continued working, not hearing the shower turn off. Your turn, she said. He glanced up. No wounds. No bruises. Nothing.