Ireland had difficulty getting their claims to independence recognized before 1921. President Wilson spoke of self determination for small nations, but he had no desire to recognize Ireland's position at the Paris Peace talks in 1919. Really up until the end of the Irish War of Independence, opinions were quite divided. Irish Diaspora communities in Australia, New Zealand, Canada, and the U.S. (along with others) attempted to get their respective governments to recognize Ireland as a sovereign state before the Treaty, but none of these governments were willing to do so. Even outside of these communities, there were activists who were appalled by the operations of the British 'Black and Tans' in 1920-1, but the various governments were still unwilling to condemn their wartime ally. The Anglo-Irish Treaty changed this situation.
After the Treaty, this situation improved vastly. The Irish Free State was recognized internationally and it developed diplomatic relations with many different nations. Essentially, the Treaty enshrined the Irish Free State as a Commonwealth territory. Article One of the 1922 Constitution stated that the Irish Free State was a "co-equal member of the Community of Nations forming the British Commonwealth of Nations." International views of the Irish Free State were analogous to views of Canada's link with Britain today: the Queen may be on the money, but the Prime Minister and parliamentary bodies call the shots. Ireland's parliament (Dáil Éireann), senate (Seanad Éireann), and its President determined policy. As Thecna2 has already noted, The Irish Free State was allowed to remain neutral during the Second World War. This is a good indicator of how their relationship functioned and the international commentators took their cues from this.
Michael Collins referred to the Treaty as a "stepping stone" to freedom, but it took a long time for the country's leaders to implement the idea of an independent republic. The Republic itself was only declared in 1948, marking Ireland's break with the Commonwealth and Britain in general. International opinion was a factor in this reluctance to declare full independence. Many nations felt Ireland was not economically stable, so they were unwilling to loan money and keep the channels of commerce open without British backing. Another reason was due to the Irish Civil War, which left some bitter political rifts in the Free State. Many Irish politicians were afraid to reopen old wounds.
Sources
Documents on Irish Foreign Policy
Foster, R. F. Modern Ireland, 1600-1972. New York, N.Y.: Penguin Books, 1989.
Kennedy, Michael. "The Irish Free State and the League of Nations, 1922-32: The Wider Implications." Irish Studies in International Affairs (1992): 9-23.
Patterson, Henry. Ireland Since 1939: The Persistence of Conflict. Dublin: Penguin Ireland, 2006.
Could I add a follow-up question. Until the Republic of Ireland Act of 1949, it appears to be unclear as to whether the head of state of Ireland was George VI or the President of Ireland. Who did other countries recognise as the head of state? Wikipedia states that:
Is this correct? How were Ireland's diplomatic relations with other countries affected by having an internationally recognised "head of state" who wasn't recognised at home? Whilst that very head of state was taking part in a world war.