I can only give an answer based on the sunni view, so anything i might say about shi'a is an assumption and should be approved or rejected by some of our shi'a bothers and sisters. Also note that -as far as i understand- this doesn't answer your question in the whole.
Why I think there's a kind of intersection is:
I have seen, read and heard a lot of similar hadith statements or fragments in Shi'a mosques or books, which are also known by sunni people.
I'll try to give a proof of that:
Hadiths are based on narrator chains, which you may find fixed up in hadith collections usually ending with the author or his student whom the book was dictated.
Now Sunni people in first place relay on a narrator chain which goes down to a sahabi (May Allah be pleased with them) who has heard or seen our Messenger (peace be upon him) doing or saying something so he was a kind of eyewitness.
Note that Shi'a reject a lot of those sahaba, but not all of them so it is not excluded at all that Shi'a have hadith narration which also exist in sunni hadith collection.
Shi'a on the other hand beside these hadith have hadiths of their Imams. Sunni wouldn't accept those as they say that nobody except the Prophets (peace be upon them) are infallible.
On the other hand historically shi'a -as it seems to me- was at first a political opposition to mainstream before becoming a theological school. Note one can't definitely say that such a definition like sunni existed before that time.
IMO the first real sect which has been formed in Islam were the Khawarij and i think their political PoV (that any Muslim under some conditions can lead the Muslim community no matter what is his sect, tribe or ancestors) is the most logical.
As sunni people even if they won't accept the shi'a Imams as infallible and won't consider a saying of them as a hadith rather then calling it athar or khabar (to make a difference between the saying/doing of the Prophet and something somebody else -who might be honored or respected- said or did). As they honor these people and even have been students of them: For example among the 4 -now known madhabs or fiqh schools- Imam Malik and abu Hanifa have been students of the Imam Ja'afar as-Sadiq and have both supported the revolt of Muhammad an-Nafs a-Zakiya against the Abbasid. So in the Muwatta' of Imam Malik you may find several ahadith he has narrated from Imam Ja'afar as-Sadiq, who basically has founded the Ja'afari fiqh school.
Note that sunni scholars have clearly admitted that those Imams on which the Shi'a relay on have been people of knowledge and honored them too as you may read in my answer here. So there's no big doubt that sunni scholars have collected some of the ahadith which we find in their hadith collections from them.
If you made a comparison between the 4 major books the Shi'a relay on and the 6-9 books of the sunni's you may find that sunni's may even have at least an equal amount of ahadith from ahl-al-bayt when it comes to connected chains (hadith musnad).
On the other hand we must confess that generally a narrator chain of a sahih hadith according sunni shouldn't contain a narrator who is well known to be from shi'a nevertheless you may find even in the sahih books of al-Bukhari and Muslim narrators who have been known shi'a or as shi'a sympathizers, if they do not offend the sahaba etc.
Honestly I don't know whether shi'a accept a hadith were you may find a sunni or sunni sympathizer in the narrator chain.
From this we can conclude that there is a common (maybe very small) set of sources when it comes to hadith. As even Sunni scholars used some statements from sahaba and Shi'a Imams as a source of jurisprudence (An example of that can be found in my answer here, where Imam Muhammad ibn al-Hassan a-Shaybani -the student of Imam abu Hanifa- has quoted a saying of Imam abu Ja'afar al Baqir).
Logically, how can Muslims rely on Hadith, when there are two different sources?
As stated in my comment shi'a and sunni claim being right, so both use the sources they relay on. Note that even among sunni schools there are major differences in the sources: So you may find different opinions on what is called Ijma', on the acceptance of customs, on the position of hadith al-ahaad, on the use of qiyas etc. But they agree that Quran and Sahih Hadith (based on the understanding of each madhab/school) are the major sources of jurisprudence.
I hope this could clear some of the confusion.