Washington — Joseph Sisco, former US under secretary of state, is that rare foreign affairs expert who also qualifies as a keen observer of US public opinion. His assessment of the President's Mideast initiative after several days of letting the dust settle encompasses both foreign and domestic arenas.
* United States public reaction:
''Broadly supportive. Also Congress will be broadly supportive.
''Of major significance is the political dialogue taking place in Israel itself, between Begin-Sharon and Peres.
''There is a broad consensus in Israel that Jerusalem remain the capital of Israel and that there not be a separate, independent Palestinian state. But there is not a consensus among Israelis for the annexation of the West Bank and Gaza.
''The American people, including the members of the Jewish community in the United States, are quite aware of the dialogue taking place in Israel and of Israeli attitudes on annexation - and this of itself has a bearing on Jewish attitudes toward the Reagan initiative.''
* Reaction from the Mideast:
''Begin's reaction, including the decision to go ahead with new settlements, is difficult but predictable. This whole thing must cook awhile.
''We must remember that this is just the beginning of a process. There is something in this proposal for everyone - something that each side can agree with and something it can disagree with.
''The Arabs have reacted predictably - with caution. But what is important is that neither side has rejected the process. In this there is hope.''
* The significance of the Reagan ini-tiative:
''It puts forward a concerted, substantive proposal for invigorating the peace process, and it foreshadows a dynamic intention on the part of the US to present ideas that go beyond a low-key, procedural role of passing messages behind the scenes.
''It is a major, substantive initiative - representing the first time this administration has defined its position substantively.
''I wish it had been done earlier, but the timing is very good right now in the aftermath of Lebanon.''
* Presidential resolve in carrying through:
''It is all-important for the United States to maintain its resolve while being flexible in detail. It is very important that we follow this approach.
''I think the President, as evidenced by Shultz's restrained, quiet response to the new-settlements decision, is indeed following this course.
''Moreover, the administration is not falling into the simplistic trap of threatening cutoff of economic and military aid. Rather, it is using the effective argument that the force of the peace process will prevail.''
* What lies ahead:
''Secretary of State Shultz will pursue explorations himself with the respective foreign ministers in late September when he goes to the UN General Assembly in New York.
''In the short run the initiative will be pursued within the quiet confines of diplomatic channels.
''It is not likely that an overt intensive effort will become manifest until the administration has closed the gap between the two sides.
''But what is of decisive importance is that the United States has made it very clear as to the substantive direction in which negotiations should go and that while it has given something to each side, it has also said 'no' to an independent Palestinian state and to Israeli annexation of the West Bank and Gaza.''
* On the possibility of a breakdown:
''Neither side will take the onus for halting the peace process. Both will seek to pressure the United States in the direction of their positions. That is what we will see in the coming months. The peace process in the Middle East is irreversible.''