Using someone else’s network safely on my iPhone

Your roommate cannot see what you are doing or access anything on your phone (or computer), just as you cannot see what your roommate is doing or access any of the data on his phone (or computer).

Most routers will provide a browsing history though, so someone who is an administrator for the router would likely be able to see which sites you have visited.

If the WiFi belongs to an Institution or Organisation then never visit sites that you do not want others to know or not legal. It is manned and monitored closely.

On your roommate's WiFI: It depends on the Admin of the Router. If your roommate is an admin and they know how to configure the router then they will know every site that you visited, the amount of data downloaded and uploaded etc..

Even if you clear history on your device the log remains on the router and your activities can be tracked.

See the pic below of My router which I have configured with whatever little knowledge that I have

You should see a list of websites (shown as IP addresses) that your router has visited. If you want to look at a specific device's activity, you ...

How to Check Browsing History on a WiFi Router: 5 Steps

Refer: Another method you can use to monitor websites visited on your WiFi is to change your router's DNS settings to point to a content filter. This ...

How to Monitor Websites that were Visited on Your WiFi

As long as your roommate is OK with you using his Wifi, he can't see what you're doing on your phone. It doesn't work that way.