Is data connected to the sim card? (Verizon)?

I'm planning to travel to China and replace the sim card on an IPhone 5s with a local one, and I want to know once I do this, will the phone only use data from the local sim card?
e.g. Is a certain carrier's data only associated with the sim card and once you take it out, you can't use their data anymore?

No, anything that that has do with data and streaming is cellular. Meaning it is from towers and stuff. What's on the SIM card is photos contacts etc. However, you can since your contacts and save your photos onto a USB of something.