In countries where armigers (persons or corporations bearing a coat of arms) must prove their entitlement to such (eg. most legal traditions within the UK, Canada, and other Commonwealth realms) what laws govern armigers bearing foreign arms (eg. a German, French, or American armiger in the UK)?
I am not sure 'laws' is the appropriate term: though the Court of Chivalry still exists it has sat only once in the last 300 years (to determine whether a theatre had the right to display the municipal arms). The decision as to who is entitled to display a coat of arms is made by the heraldic authority in each country (the College of Arms for England and Wales and much of the Commonwealth; Lyon King of Arms for Scotland). These authorities work closely with each other (unsurprisingly when you consider that one of their original functions was to broker peace treaties), so it is likely that if arms granted in one country will be recognised by the heralds in another; but the decision is made individually.
I found one interesting quote, though my Google-fu does not allow me to identify the source:
M . J. Sayer writes some interesting things about registrations of foreign arms with the College of Arms in his "English Nobility: The Gentry, the Heralds and the Continental Context" (Norfolk Heraldry Society, 1979). On pages 17-18 Sayer writes "where foreign arms are recognized in England, they of course rank in England as ensigns of nobility, even if the family was not noble abroad, an anomaly reflecting the greater success of the English crown's control over arms." A footnote continues: "Sir Anthony Wagner kindly informs me that 'foreign arms for which the authority of a document from a Sovereign or heraldic authority can be produced have a head start towards recognition here over Burgher arms. Nevertheless, though acceptance of the document may be the first stage, it has often been thought necessary that there should be a second stage of confirmation of the right to use the arms in this country. One sees that there could in theory, and sometimes in practice, be a clash with the design of existing English arms. At the present day we keep a record entitled Foreign Arms where such documents are entered when accepted as valid, but over and above this there needs to be acceptance by Patent or otherwise for use in England.' (30 March 1978)...Grants by foreign authorities to persons not within their jurisdictions, i.e. to nationals of a third country, are not accepted for registration."
The College of Arms has at times contacted foreign governments to ascertain the status of their native heraldic authorities.
Finally, I am surprised by your reference to American armigers; it is my understanding that the Article I of the US Constitution forbids the states to create titles of nobility.