In order to achieve this communication i suggest that the child (CardSide Component) communicates with the Card Component via Events .
so when the user finish his operation on the card component an event is fired passing all the data to the parent let me show you an example for what i mean : 
Card Component
class Card extends Component {
  handleCompelete = data => {
    //the data here are all the data entered from the child component
    //do some sorting using table name
  };
  render() {
    return <CardSide onCompelete={this.handleCompelete} />;
  }
}
CardSide Component
class CardComponent extends Component {
  render() {
    return (
      <div>
         {/* data here reprensets what you need to transfer to parent component */}
        <button onClick={() => this.props.onCompelete(data)} />
      </div>
    );
  }
}
Edit
You cannot access the state of the child component as it is private to it.
Regarding to the props , you can access it but it is ReadOnly that is passed from the parent component but the child component cannot modify it .
Actually there is a way to access the component children (but i see it will complicate your code rather than simplifying it and i do not recommend this )
lets say that this is you app.js 
class App extends Component {
  constructor() {
    super();
    this.state = {
      name: "React"
    };
  }
  render() {
    return (
      <div>
        <h1>Title</h1>
        <Card>
          <CardSide someProp="My Child Prop Value" />
        </Card>
      </div>
    );
  }
}
as you can see i included CardSide with property with name someProp as a child for Card rathar than inserting it inside Card Component 
In the Card Component i accessed the children property as the following :
class Card extends Component {
  handleCompelete = data => {
    //the data here are all the data entered from the child component
    //do some sorting using table name
  };
  render() {
    return <div>
     {this.props.children}
     {console.log(this.props.children)}
     {this.props.children.props.someProp}
    </div>;
  }
}
and the CardSide Component 
class CardSide extends Component {
  render() {
    return (
      <div>
         {/* data here reprensets what you need to transfer to parent component */}
        <button onClick={() => this.props.onCompelete(data)} >
        Hello btn
        </button>
      </div>
    );
  }
}
As you can see it will get your structure more complicated and it will be hard to know who is the children for the card component without intensive tracing .
you can see the code in action via this link https://stackblitz.com/edit/react-fxuufw?file=CardSide.jsx