The phone does not record incoming messages. Do I have to subscribe to the phone company's answering sevice in order to get incoming messages?

You may be subscribed to a voicemail system provided your telephone company. This system records messages when people call while you're already on the phone. Please contact your telephone company for instructions on accessing their voicemail system.

Make sure it's in the "Answer" mode, not the "Announce" mode. The Announce mode merely plays an outgoing message but does not record an incoming message. The Answer mode does record incoming messages.
Since you declined to specify the model of your Uniden phone I can't tell you exactly how to switch modes. Select the model of your phone from the list beginning here to download a copy of the manual.

The Uniden D1660-2 isn't an answering machine. If you have a separate answering machine connected to your phone line, please consult its documentation. If you subscribe to a voicemail system offered by your phone company, please contact them for instructions on accessing their voicemail system.
If you need a D1660-2 manual, you may download a copy here.

The 6041 isn't an answering machine. If you have a separate answering machine connected to your phone, consult its documentation. If you subscribe to a voicemail service provided by your telephone company, contact them for instructions.

You can download a copy of the manual from the manufacturer's web site at
http://vt.vtp-media.com/ecp/documents/product_Product/85/UserManual/245/6041_Manual.pdf

The CS6114 isn't an answering machine. If you have a separate answering machine connected to your phone line, consult its documentation. If you subscribe to a voicemail system offered by your phone company, contact them for information and help.

You're subscribed to a voicemail system provided by your telephone company. This system records messages when people call while you're already on the phone. Please contact your telephone company for instructions on accessing their voicemail system.

MAKE SURE YOU DONT HAVE VOICEMAIL SERVICE THROUGH YOUR PHONE COMPANY SEVICE PROVIDER, ITS POSSIBLE THAT MIGHT BE RETRIEVING YOUR MESSAGES.

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For voice mail service subscribers
If you subscribe to a flat-rate service
package that includes Caller ID, call
waiting, voice mail, and unlimited
local/regional/long distance calls, please
note the following:
L To use the voice mail service provided
by your service provider/telephone
company rather than the unit's
answering system, turn off the
answering system (page 30).
L To use this unit's answering system
rather than the voice mail service
provided by your service provider/
telephone company, please contact
your service provider/telephone
company to deactivate your voice mail
service.
If your service provider/telephone
company cannot do this:
- Set this unit's "Ring count"
setting so that this unit's answering
system answers calls before the
voice mail service of your service
provider/telephone company does.
It is necessary to check the number
of rings required to activate the
voice mail service provided by your
service provider/telephone
company before changing this
setting.
- Change the number of rings of the
voice mail service so that the
answering system can answer the
call first. To do so, contact your
service provider/telephone
company.

You're subscribed to a voicemail system provided by your telephone company that records messages when someone calls while you're already on the phone. Please contact your telephone company for instructions on accessing their voicemail system.