I do not know about its correctness, or faithfulness to Arabic or Hebrew, or (if it is not so faithful) its origin, but I think the pronunciation described in the OP - something like sign-yai - is common in Britain. Indeed, it how I would probably say it myself. I do know that the BBC is traditionally very careful about pronunciations of foreign words, and I probably learned my pronunciation from BBC newsreaders. I sort of assumed that it was because the n was to be pronounced as if it had a tilde. (I don’t think I have ever heard “sigh-a-nie”.)
Dictionary.com gives the following pronunciation options: “sahy-nahy, sahy-nee-ahy” or, in IPA “saɪ naɪ, saɪ niˌaɪ”. I don’t read IPA, and their non-IPA version does not make much sense to me, but, either way, it appears that saying it either with or without the extra vowel after the n is established English usage.