What is the relationship between the letters samekh and sin? Did they ever have distinct sounds? Why do they exist as separate letters?
"A sin is just a samech with three branches."
-- A contemporary American ראש ישיבה
The idea here is that in modern usage they are both actually interchangeable with samech.
For an illustration of this interchangeability see the 15th line of the alphabetical acrostic א-ל אדון, in which a sin appears where we would expect a samech, or the common root א.ר.ס used by חז"ל in place of the Torah's synonymous root א.ר.ש.
The difference in sound between the samech, a voiceless alveolar fricative ([s]) and sin, a putative voiceless lateral fricative ([ɬ]), is hypothesized to have been lost as early as Biblical Hebrew, with its remnants still evident in transliterations like "Chaldean" for "כשדי" and "balsam" for "בושם". In each case, the sin that we would conventionally pronounce as [s] was ostensibly closer to [ɬ], which is, in a way, halfway between /s/ and /l/.
There is a complicated diachronic explanation of exactly how and why this shift took place, which is summarized very briefly on page 73 of this paper, by Biblical Hebraicist Gary Rendsburg.
I am not a speaker of modern Hebrew, but am a beginning level learner of Biblical Hebrew. However I did study about ancient Hebrew pronunciation, linguistics, and specifically phonetics quite a bit. I lean heavily in three areas, 1. "logical" phonetics, belief that original/Biblical Hebrew is designed and there should be logical rules to discover when pursuing such questions. 2. Phonotactics constrains languages and thus Semitic languages can be helpful with similar structures phonetically 3. Mizrahi Hebrew class of Hebrew pronunciation, as these are closest to the phonetic logic of the Hebrew Bible. Over the years of pursuing answers to Hebrew pronunciation questions, I came across an example of a variety of Yemenite Hebrew (lost at this time to myself) said to pronounce Schin with dot to the upper left as IPA ɕ, which is in the Polish language, for example, as ś. To me this is a logical progression of the tongue contact with the roof of the mouth front to back for each sibilant, instead of a "radical" departure for some sort of lateral. It sounds nice, is logical, flows easily in the mouth, but I sure would like to find examples in Mizrahi Hebrew/Semetic languages of IPA ɕ, beyond the reference I lost. For now I use it in my studying, and it makes a great distinction for disambiguating biblical/Hebrew words by sound.
Being able to read and speak Hebrew fluently,I can answer this question. Both Samekh and Sin make an 's' sound. You can tell what sound a Hebrew letter makes by the first sound you speak when pronouncing the it. For example, Bet makes a 'b' sound. However, this technique will not work for the Hebrew letters Alef, Ayin, Chet and chaf. Alef and Ayin being silent and there being no English equivalent for Chet and Chaf is the reason for this.