Short answer: in daily life Panama is generally safe for gays, lesbians and transgender people. Superficially, everyone seems to get along.
Long answer: Being gay and talking about acceptance and discrimination is a huge taboo... You can really only feel and notice that if you are member of the LGBTI-community... I'm gay and run an LGBTI- organization, called: "Unión de la diversidad", so we have quite an idea about what's going on around here. Although people do not really get harassed on the streets, over 80% (Pew research, 2014) is against gay-partnership. Hate-speech is very common (especially online), see some articles on "The Panama Observer" (linking is not allowed so I can't send the link here).
Police do abuse gay people if they feel like it. We have a file with many reported cases that are not solved or looked into.
Some people do lose their jobs is if they openly come out of gay. It is our estimation that 60-80% are in fact in the closet. We have had cases of people who were thrown out of their parent's house and are forced to live on the streets. I know lawyers, doctors, business owners and politicians, that have to live inside the famous "closet," because they are afraid for their careers (and with reason).
There is enormous stigmatization going on here; gays are generally seen as feminine / hairdressers / make-up artists, but any other type of career will be very hard for an openly gay or lesbian person. We offer a weekly support group for LGBTI (first in Panama) and talk a lot about these topics; it's shocking to hear and see what comes up. Many people confuse being gay with pedophilia or zoofilia, which is very painful for us as gays.
Some real estate ads will say: "No gays"... Just this week we started a case against someone regarding that advertisement. This happens frequently.
The gay pride; it seems like this is all going smoothly, but this year the stage was ruined right before the parade. Possibly this was directed against the gay-community. I've walked with the gay pride and there are many people looking at the parade and shouting vulgarities such as "dirty gays," "You are all going to hell" This is something I have never experienced anywhere else, but then again, I mostly traveled in Europe.... My advice is to be careful, live in an upscale area, pick good friends and keep your expectations regarding acceptance very low.
(Rainbow Hammock hand made in Panama, pictured.)