Genuine weaknesses of the GDPR
Quite apart from the right of access under Article 15, Article 14 requires a data controller obtaining personal information from sources other than the data subject to inform the data subject of the processing and their rights (as under Article 13 where the data subject supplies the information) and also of where the data were obtained from. So far all well and good, as the data subject should in principle have been informed, either by the data controller they provided their data to (under Article 13) or the recipient of a data sharing (under Article 14).
However Article 14.5(b) provides a discretionary get out clause (that I guess most behemoth data slurpers might choose to rely on) if "the provision of such information proves impossible or would involve a disproportionate effort".
Furthermore, it appears so far (there being very little precedent yet) that where a data controller shares personal data with a third party data controller on the basis of legitimate interest, the responsibility of the sourcing data controller is limited to the actual process of sharing (as a joint controller for that process) unless the sharing involves a "transfer" to a third country. Otherwise, the sourcing data controller is not responsible even for checking whether the recipient processes the data in accordance with the legislation.
Consequently, you've asked the $64,000 question. How indeed?
The ideal answer would be enforcement of Article 14 with strict attention to abuse of 14.5(b) to facilitate evasion. Given an enforcement regime that essentially relies on complaints (policing by data subjects) that's not likely to happen soon.
The position is in principle different if the third party is a data processor for the sourcing data controller, in which case the obligations are well defined. However even in that case two major problems have not yet been solved:
[1] Most of the behemoths that provide "processing" for data controllers under the GDPR nevertheless impose their own unilaterally defined non-negotiable contracts on the data controller. This inverts the status of the controller/processor relationship and should in principle be unlawful, but has not yet gained sufficient attention;
[2] Many of the behemoths providing "processor" services currently include in their privacy statements to data subjects AN assumed right also to act as data controllers for for their own purposes of the information provided to them in their capacity as processors. Whether this could be considered unlawful is still an open question, as the lawful basis usually relied on is the much abused "legitimate interest".
The greatest weakness of the GDPR is that it has not been in force for long enough. It is likely to take many years of precedent before all these issues are considered properly and ruled upon conclusively.