Do the people in the matrix know about the Agents, and their abilities?

In the first Matrix movie, after the fight in the subway station, Neo is running around in the city, and Agents leap from body to body to try to stop him. Do people surrounding the people taken over notice that it was take over?

In one part of the persecution, Neo gets into a market, and we see an Agent take over a woman that has a child holding her hand. She turns into the Agent and starts shooting and the child is there watching all of that, does the child remembers that?

Answer

There are two main elements at work here, both relating to memory manipulation;

• When an agent occupies a bluepill (someone still connected to the Matrix), that person clearly has no memory of what has happens. They simply fugue until they're released from control. This can be seen in the Webcomic "Day in, Day out", where a character retains the injuries her Agent/occupier incurred, but none of the memories of how she got them.

• It's pretty clear from the film and supporting media that the machines are capable of altering memories. Not only are they able to do this on a grand scale, across the entire population of the Matrix when they periodically reset the Matrix to an earlier time period (something which would presumably require them to create 6 billion separate complete sets of memories and implant them simultaneously), but they can also mess with memories in a very specific and individual way, as evidenced by Cypher stating that he would like his memories removed as part of the deal to reinsert him into the Matrix.

Cypher : I don't want to remember nothing. Nothing. You understand?

At no point do we observe the Machines lying or dissembling and it's reasonable to take them at face value that they can do this.

• Finally, it stands to reason that anyone who's seen an Agent transform is liable to simply believe they've suffered from an hallucination and act accordingly by seeking professional help. In "A path among the Stones", we see a young girl who's seen the truth of the Matrix. When this is reported to her psychologist, it triggers a groups of Agents turning up to take her away, presumably to either kill her or wipe her memory.

Of course, if the experience of seeing (or being occupied by) an Agent convinces the victim to dig deeper they may eventually find out the truth, that the world around them is far stranger than they first thought.