I sent this email to Skype Support:
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Hi. I'm having a problem and would like some help.

I have Skype set up on my MacBook and have enabled voicemail.

I have Skype installed on my iPhone also.

Voicemail works if:
A. Skype on MacBook is logged in.
B. Skype on MacBook and on iPhone are logged in.
C. Skype on MacBook and on iPhone are NOT logged in.

Voicemail does not work if:
C. Only Skype on iPhone is logged in.

Please advise.
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I got this reply:
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Thank you for contacting Skype Support.

We are sorry to hear that you are having problems with Skype.

We are aware of this problem and are working hard on fixing it as soon
as possible. We apologize for the inconvenience that this may cause.

Please let us know if you have any other concerns or questions, and we
will be glad to assist you.

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I'm surprised that this issue isn't discussed any where in this forum (that I could find). Clearly I'm not the only one with this problem since Skype is aware of the problem.

Support was unable to provide any information on releasing an update that addresses the problem. Voicemail is an important service to me so I really hope that the development team will make fixing it a priority.