Code snippets from: http://adtennant.co.uk/blog/2014/03/20/component-controller-communication-using-interfaces-part-1/

gistfile1.cls

public class CounterComponentController {
    public CounterComponentListener listener { get; set; }

    private Integer count;

    public CounterComponentController() {
        count = 0;
    }

    public Integer getCount() {
        return count;
    }

    public PageReference increment() {
        count++;

        if(listener != null) {
            listener.onChange(count);
        }

        return null;
    }
}

gistfile2.html

<apex:component controller="CounterComponentController">
    <apex:attribute name="componentListener"
        description="The listener for this component."
        type="CounterComponentListener"
        assignTo="{!listener}"
    />
    <apex:outputPanel id="output">
        <apex:outputText value="{!count}"/>
    </apex:outputPanel>
    <apex:form>
        <apex:commandButton action="{!increment}" value="Increment"/>
    </apex:form>
</apex:component>

gistfile3.cls

public interface CounterComponentListener {
    void onChange(Integer newValue);
}

gistfile4.html

<apex:page controller="BasicCounterPageController">
    <apex:pageMessage severity="warning" strength="3" summary="It's over 2!" rendered="{!ShowAlert}"/>
    <c:CounterComponent componentListener="{!ComponentListener}"/>
</apex:page>

gistfile5.cls

public class BasicCounterPageController implements CounterComponentListener {
    private Boolean showAlert;
 
    public BasicCounterPageController() {
        showAlert = false;
    }
 
    public Boolean getShowAlert() {
        return showAlert;
    }
 
    public CounterComponentListener getComponentListener() {
        return this;
    }
 
    public void onChange(Integer newValue) {
        showAlert = (newValue > 2);
    }
}