While it is my responsibility to review and respond to documents filed in a case, the client has an interest in reading the documents also and I provide them to the client for their review and comment. It is important, too, that the response reflects the position of the client, and it would be incorrect to respond without the input of the client.
The answer will vary by law firm and by client. Some clients want to be involved, others don't. Most procedural matters are not sent to the client, but most substantive matters are, and all settlement communications must be sent to the client. Not reviewing every document is not the same as the client not remaining in good contact with his legal counsel, for clarifications and a number of other issues.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
It indeed varies from case to case. One thing is for sure, as the client, if you want to be copied on everything going on in the case, then you have that right. Many lawyers will talk with the client regularly to give updates and at every important pre-trial phase of the case to inform them on the main issues and the results of any depositions that are taken. This is the course I take, unless the client asks for specific documents or information. If you wish to be provided with more information on the case, make an appointment with your lawyer to review the file and ask questions.