The Office of Strategic Services, which was created specifically to handle the wartime spying necessary during WWII, ended up becoming a very large organization as the war progressed. Estimates vary, but it's likely that it was employing somewhere between 12,000 to 24,000 people at its peak operational period. In 2008, the records for all OSS agents were released, which revealed that there were plenty of foreigners/foreign-born individuals on the OSS dole working for the organization. These ranged from soldiers to political dissidents to scientists or linguists. (As an aside, OSS records are great, they employed everyone from professional wrestlers and baseball players to Julia freakin' Child). So, in answer to your first question, yes, it was very common for the OSS to pick up foreign nationals as operatives, including those from Germany or Eastern Europe. If anything, German-born operatives were particularly prized because it was far easier for them to infiltrate Germany than an American or another European. Eventually, the OSS even started a program to recruit German POWs for spying missions within Germany.

So, that leads to your next question, how could the OSS trust all these foreign nationals not to betray them? Well, the OSS did not pick up just any foreign national to work for them. In the case of recruiting German POWs, for example, OSS Secret Intelligence Chief Harry Hyde recruited a team of trusted military officers and civilian experts to perform extensive interviews and background checks on all potential POW recruits. This included psychological evaluations, extensive secret observations of behavior both in and outside of the interview room, and of course, the interviews themselves. Only once it had been determined that a potential recruit was completely trustworthy and reliable would they actually begin training.

As for regular foreign national recruits, the process was actually not that different. Potential recruits, at least at first, were usually brought to a designated secret location with similar novices for an evaluation period usually lasting a weekend. During that time, attendees were given strict instructions not to reveal their true names and underwent a battery of psychological evaluations, personal history discussions, personality analysis, interviews, and observation during tasks. The evaluators on hand kept detailed records of the performance of candidates during this time period and those who were deemed incapable or untrustworthy were simply let go without ever having gotten too deep into the organization.

Obviously, the system was not perfect, but in general it was an effective form of screening all potential operatives to ensure their loyalty and commitment to the cause. As such, anyone capable of completing the evaluation process was generally deemed to be trusted for secret work, though differing scores during the evaluation could determine just how much secret material or classified work an individual actually got. Of course, not all recruits underwent that process either, but many of them did. Furthermore, any additional problems with loyalty would most likely have come out during the training process following the evaluation period, so there was one additional failsafe should the initial evaluation not work.

Some general sources from the internet:

http://warfarehistorynetwork.com/daily/wwii/oss-in-germany/

http://people.wku.edu/sally.kuhlenschmidt/whimsy/oss/oss.htm