I wonder why Mallory ran into the kitchen to answer the phone instead of answering it in the living room.

I think the reason she runs to the kitchen phone is because that's where the family usually is, and those scenes usually involve interraction between her and the family. If she were to answer the phone in the living room, while her family was in the kitchen it would be hard to do the scene. I think from the writers, and directors perspective, that is why she does this. I think an explanation as to why she does this from a characters stand point is number one, the character of Mallory is supposedly a moron, so it makes sense that she would answer the kitchen phone while she's in the living room. Another explanaion could be that she's in the hallway when she comes running in, and is actually closer to the kitchen. Those would be my guesses as to why she does this.

She could have possibly wanted her privacy when she talked on the phone.

If she wanted her privacy, I don't think she would have answered in the kitchen where her whole family was. If that were the case she most definitely would have answered in the living room, or upstairs. Her conversations usually seemed to be about clothes, or the mall, it didn't really seem to involve much personal stuff, so I don't believe privacy was a high priority for her.