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If I ask a sell-side desk "A" for a "valuation" of a relatively simple OTC product (equity derivative, or 1st generation equity exotic), what are the reasons/main reason why a different sell-side desk "B" (at some other bank) would arrive at a different valuation for exactly the same product?

LaplaceKis
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Pricing these products is subject to different models. One bank might calibrate their vol surfaces slightly differently and have different skews on their respective smiles. Also, one bank might assume a slightly different stochastic process in their model for the pricing - or use the same model, but calibrated with different parameters.

Lastly, If I already am very very long a lot of SPX gamma, I probably don't want to buy a short dated digital near the money from you - I would probably prefer to sell it to you to unload some gamma - and I will show you prices according to my preference - of course if you buy from me immediately, no questions asked, I may suspect that I was a bit low and move the price up a bit afterwards.