For a User Initiated Send (EmailSendDefinition object), you can retrieve the SendDefinitionList property, and if there are multiple lists/DEs associated with the send, it will show them to you... (More below).
If you want to see the DE associated with Triggered Sends, there's not currently a way to retrieve that via the API :( I had SFMC Data Support create a custom extract for my account (something like Populate TriggeredSend To DE Match), that runs daily and stores the correlation between Data Extension and Triggered Send Definition in a shared DE that can be accessed by any BU to get that info. Then you'd just retrieve the DE Name from the extract target DE for a given TSD's customerkey.
To retrieve the lists/DEs used in a User Initiated Send, you'd use this SOAP call (again, relevant property being SendDefinitionList):
<soapenv:Envelope xmlns:soapenv="http://schemas.xmlsoap.org/soap/envelope/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:wsa="http://schemas.xmlsoap.org/ws/2004/08/addressing" xmlns:wsse="http://docs.oasis-open.org/wss/2004/01/oasis-200401-wss-wssecurity-secext-1.0.xsd" xmlns:wsu="http://docs.oasis-open.org/wss/2004/01/oasis-200401-wss-wssecurity-utility-1.0.xsd">
<soapenv:Header>
<wsse:Security soapenv:mustUnderstand="1">
<wsse:UsernameToken>
<wsse:Username>${#Project#username}</wsse:Username>
<wsse:Password>${#Project#pw}</wsse:Password>
</wsse:UsernameToken>
</wsse:Security>
</soapenv:Header>
<soapenv:Body>
<RetrieveRequestMsg xmlns="http://exacttarget.com/wsdl/partnerAPI">
<RetrieveRequest>
<ClientIDs>
<ClientID>${#Project#BU}</ClientID>
</ClientIDs>
<ObjectType>EmailSendDefinition</ObjectType>
<Properties>Additional</Properties>
<Properties>AutoBccEmail</Properties>
<Properties>BccEmail</Properties>
<Properties>CategoryID</Properties>
<Properties>CCEmail</Properties>
<Properties>CreatedDate</Properties>
<Properties>CustomerKey</Properties>
<Properties>DeduplicateByEmail</Properties>
<Properties>DeliveryProfile.CustomerKey</Properties>
<Properties>Description</Properties>
<Properties>DynamicEmailSubject</Properties>
<Properties>Email.ID</Properties>
<Properties>EmailSubject</Properties>
<Properties>ExclusionFilter</Properties>
<Properties>IsMultipart</Properties>
<Properties>IsSendLogging</Properties>
<Properties>IsWrapped</Properties>
<Properties>ModifiedDate</Properties>
<Properties>Name</Properties>
<Properties>ObjectID</Properties>
<Properties>SendClassification.CustomerKey</Properties>
<Properties>SendDefinitionList</Properties>
<Properties>SenderProfile.CustomerKey</Properties>
<Properties>SendLimit</Properties>
<Properties>SendWindowClose</Properties>
<Properties>SendWindowOpen</Properties>
<Properties>SuppressTracking</Properties>
<Properties>TestEmailAddr</Properties>
<Filter xsi:type="SimpleFilterPart">
<Property>Name</Property>
<SimpleOperator>equals</SimpleOperator>
<Value>My Test UISend</Value>
</Filter>
</RetrieveRequest>
</RetrieveRequestMsg>
</soapenv:Body>
</soapenv:Envelope>
The output should look something like this:
<soap:Envelope xmlns:soap="http://schemas.xmlsoap.org/soap/envelope/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:wsa="http://schemas.xmlsoap.org/ws/2004/08/addressing" xmlns:wsse="http://docs.oasis-open.org/wss/2004/01/oasis-200401-wss-wssecurity-secext-1.0.xsd" xmlns:wsu="http://docs.oasis-open.org/wss/2004/01/oasis-200401-wss-wssecurity-utility-1.0.xsd">
<soap:Header>
<wsa:Action>RetrieveResponse</wsa:Action>
<wsa:MessageID>urn:uuid:13505862-fbdf-4457-ac1e-8f9bc173d140</wsa:MessageID>
<wsa:RelatesTo>urn:uuid:8a332506-5dc9-4044-962e-927a39316dfe</wsa:RelatesTo>
<wsa:To>http://schemas.xmlsoap.org/ws/2004/08/addressing/role/anonymous</wsa:To>
<wsse:Security>
<wsu:Timestamp wsu:Id="Timestamp-d1c7c006-05aa-4f9f-bd9d-c5e6d0dc6831">
<wsu:Created>2018-08-06T23:53:34Z</wsu:Created>
<wsu:Expires>2018-08-06T23:58:34Z</wsu:Expires>
</wsu:Timestamp>
</wsse:Security>
</soap:Header>
<soap:Body>
<RetrieveResponseMsg xmlns="http://exacttarget.com/wsdl/partnerAPI">
<OverallStatus>OK</OverallStatus>
<RequestID>e17a8f45-3f2f-4d88-b6f6-5145a1b547cc</RequestID>
<Results xsi:type="EmailSendDefinition">
<PartnerKey xsi:nil="true"/>
<CreatedDate>2017-02-21T14:52:09.503</CreatedDate>
<ModifiedDate>2017-02-21T15:12:49.33</ModifiedDate>
<ObjectID><!--OBSCURED--></ObjectID>
<CustomerKey><!--OBSCURED--></CustomerKey>
<Name>SUPPRESSION_TEST_C6012</Name>
<Description/>
<CategoryID>40106</CategoryID>
<SendClassification>
<PartnerKey xsi:nil="true"/>
<ObjectID xsi:nil="true"/>
<CustomerKey>Default Transactional</CustomerKey>
</SendClassification>
<SenderProfile>
<PartnerKey xsi:nil="true"/>
<ObjectID xsi:nil="true"/>
<CustomerKey>DynamicFromName</CustomerKey>
</SenderProfile>
<DeliveryProfile>
<PartnerKey xsi:nil="true"/>
<ObjectID xsi:nil="true"/>
<CustomerKey>Default</CustomerKey>
</DeliveryProfile>
<SuppressTracking>true</SuppressTracking>
<IsSendLogging>true</IsSendLogging>
<SendDefinitionList>
<PartnerKey xsi:nil="true"/>
<ObjectID><!--OBSCURED--></ObjectID>
<List>
<PartnerKey xsi:nil="true"/>
<ID><!--OBSCURED--></ID>
<ObjectID xsi:nil="true"/>
</List>
<SendDefinitionListType>SourceList</SendDefinitionListType>
<CustomObjectID><!--OBSCURED--></CustomObjectID>
<DataSourceTypeID>CustomObject</DataSourceTypeID>
<IsTestObject>false</IsTestObject>
<SalesForceObjectID/>
<Name/>
</SendDefinitionList>
<SendDefinitionList>
<PartnerKey xsi:nil="true"/>
<ObjectID><!--OBSCURED--></ObjectID>
<SendDefinitionListType>ExclusionList</SendDefinitionListType>
<CustomObjectID><!--OBSCURED--></CustomObjectID>
<DataSourceTypeID>CustomObject</DataSourceTypeID>
<IsTestObject>false</IsTestObject>
<SalesForceObjectID/>
<Name/>
</SendDefinitionList>
<SendDefinitionList>
<PartnerKey xsi:nil="true"/>
<ObjectID><!--OBSCURED--></ObjectID>
<SendDefinitionListType>ExclusionList</SendDefinitionListType>
<CustomObjectID><!--OBSCURED--></CustomObjectID>
<DataSourceTypeID>CustomObject</DataSourceTypeID>
<IsTestObject>false</IsTestObject>
<SalesForceObjectID/>
<Name/>
</SendDefinitionList>
<Email>
<PartnerKey xsi:nil="true"/>
<ID><!--OBSCURED--></ID>
<ObjectID xsi:nil="true"/>
</Email>
<BccEmail/>
<AutoBccEmail/>
<TestEmailAddr/>
<EmailSubject>%%= v(@subjectline) =%%</EmailSubject>
<DynamicEmailSubject>%%= v(@subjectline) =%%</DynamicEmailSubject>
<IsMultipart>true</IsMultipart>
<IsWrapped>true</IsWrapped>
<DeduplicateByEmail>false</DeduplicateByEmail>
<ExclusionFilter/>
<Additional/>
<CCEmail/>
</Results>
</RetrieveResponseMsg>
</soap:Body>
</soap:Envelope>
Notice, there are multiple <SendDefinitionList>
elements, and they have different SendDefinitionListTypes, depending on how they are used.