I created a sample application for the scenario and it is not necessary for the consuming client to include a certificate (given configurations similar to below are used)
WCF service with Transport security only
Bindings
<wsHttpBinding>
<binding name="wsHttpBindingConfiguration" receiveTimeout="00:10:00" sendTimeout="10.00:00:00" maxBufferPoolSize="1073741824" maxReceivedMessageSize="1073741824">
<readerQuotas maxDepth="32" maxStringContentLength="2147483647" maxArrayLength="16384" maxBytesPerRead="4096" maxNameTableCharCount="16384"/>
<security mode="Transport">
<transport clientCredentialType="None"></transport>
</security>
</binding>
</wsHttpBinding>
The configuration of the service endpoints
<service behaviorConfiguration="noClientCertBehavior" name="WCFCallbackTry.Service1">
<endpoint address="https://machineName:8056/Service1.svc" bindingConfiguration="wsHttpBindingConfiguration" binding="wsHttpBinding"
contract="WCFCallbackTry.IService" name="HttpsEndPoint" />
<endpoint address="mex" binding="mexHttpsBinding" contract="IMetadataExchange" />
<host>
<baseAddresses>
<add baseAddress="https://machineName:8056/Service1.svc"/>
</baseAddresses>
</host>
</service>
Service Behavior
<behavior name="noClientCertBehavior">
<serviceDebug includeExceptionDetailInFaults="true"/>
<serviceMetadata httpsGetEnabled="true"/>
<serviceCredentials>
<serviceCertificate findValue="9d4c41cde9d2b82d751a1234fd2eb6df98d3b576" storeLocation="LocalMachine" storeName="My" x509FindType="FindByThumbprint"/>
</serviceCredentials>
</behavior>
Client
Bindings and endpoint
<system.serviceModel>
<bindings>
<wsHttpBinding>
<binding name="HttpsEndPoint">
<security mode="Transport">
<transport clientCredentialType="None" />
</security>
</binding>
</wsHttpBinding>
</bindings>
<client>
<endpoint address="https://machineName:8056/Service1.svc" binding="wsHttpBinding"
bindingConfiguration="HttpsEndPoint" contract="ServiceReference1.IService"
name="HttpsEndPoint" />
</client>
Also refer link for more information on different configurations
Note:Client and Service reside on same machine