When should the stakeholders be involved in a project's communication?

Stakeholders should be involved in a project from he beginning. That is depending on the stakeholders they would need to be involved from the initiation phase.

At each and every step of the project as "its their project not yours" , so they are the ones who should be informed first about everything related to the project whether its time, cost , scope , quality or resource.

Ideally all stakeholders should be INFORMED at every stage.

But practically It depends on type of stakeholder. Type of stakeholder is defined by "interest of stakeholder" and "Power of stakeholder".

Degree of interest and power of stakeholders defines what their impact will be on project and how project will effect them.

More powerful and interested stakeholders should be informed at every stage of project.

Early and Often - Stakeholders are primarily concerned with Vision, Schedule, Budget and Scope.  They may also be involved in Political, Legal ramifications and Public Perception issues. 

They should be consulted early and should be kept apprised of any issues that may affect the above. 

The sooner, the better.. Planning phase includes Stakeholders mgmt. plan for a reason.

The voice of stakeholders of getting involve on project communications starts from defining, initiating and drafting the project business case till moving forward to project milestone i.e. POC, execution, completion/handover and after the project team sign off till end of project closure.

By communicating with stakeholders early and frequently, you can ensure that they fully understand what you are doing and understand the benefits of your project – this means they can support you actively when necessary.  Not only they will support you, their input can also improve the quality of your project.

Stakeholders should be involved at every stage.  By definition, the project affects them, so they care about the output and have useful information for the input.