Amino acids side effects can be severe. It is often said that amino acids are safe and natural, and this is true if they are used properly. But unfortunately there is very little reliable testing available to assess which amino acids you need more of. That means it is easy to have too much either in one dose or over a period of time. From dietary sources alone it is virtually unheard of to have too much of an amino acid and to suffer side-effects.
Tryptophan (commonly used for anxiety and depression) can have side-effects at relatively low doses. Side-effects can include nausea and dizziness and, in cases of large doses, can result in mania and seizures. It should not be taken alongside other drugs without medical supervision.
Histidine can cause a rise in histamine and make allergies and asthma worse. Large doses should be avoided and caution taken. Histidine is said to be effective for anxiety in some cases
Tyrosine (indicated for anxiety) is involved in the production of dopamine and epinephrine and to some extent serotonin and is favored by some as a treatment for depression. But too much can cause over-stimulation including anxiety, panic attacks and insomnia. Use with care and start at low doses. Read this post: Tyrosine for Anxiety.
Glycine can give palpitations and panic attacks to people who are sensitive if they take too much.
Glutamine can cause mild stomach upsets but is not known for severe side-effects at any ordinary dose.
Taurine is not known to cause serious side-effects at this time, although there are isolated incidents of some negative outcomes that may or may not have been related to large taurine intake.