The short answer to the question is yes, they do;  the longer answer to the question is that their rights to first amendment protection of speech is very limited, and litigation over those rights will be stringently examined, or better put, subject to strict scrutiny.  In Ruotolo v Mussman & Northey
2012 NY Slip Op 30860(U)  April 3, 2012  Sup Ct, NY County  Docket Number: 109449/2008
Judge: Saliann Scarpulla both plaintiff’s case against the Police Department and his later legal malpractice case against the attorneys are lost.

"Plaintiff Angelo Ruotolo (“Ruotolo”) is a former New York City Police Department (“NYPD”) officer. In June 2003, while still employed with the NYPD, Ruotolo commenced a civil rights action in the United States District Court for the Southern District of New York (“the civil rights action”) against the City of New York and several NYPD supervisors (collectively “the City defendants”). Non-party William Rold (“Rold”) initially represented Ruotolo in the civil rights action. In the civil rights action Ruotolo alleged that the City defendants retaliated against him for writing a report (“the Report”) about the possible health effects of environmental contamination in the 50th precinct, where Ruotolo served as a Command Safety Officer. Ruotolo alleged that after writing the Report, the City defendants arbitrarily denied him time off and overtime, reassigned him repeatedly, and disciplined him for trivial reasons. Ruotolo asserted claims that these retaliation-or.y actions violated the First Amendment and Due Process clauses of the U.S. Constitution, and various state whistle blower laws. "

"In November 2003, the City defendants moved to dismiss Ruotolo’s complaint, arguing that the Report was not protected by the First Amendment because Ruotolo prepared it in his capacity as a public employee, not as a private citizen. On August 25, 2004, Judge Stein denied the City’s motion with respect to the First Amendment and Due Process claims but granted the motion with respect to the state whistle blower claims. Then, in February 2004, the NYPD charged Ruotolo with visiting an out of borough location while on duty, subsequently placed him on modified duty and confiscated his firearms. Ruotolo eventually retired from the NYPD without a permit to carry a firearm as a civilian. Also, by mid-2004 conflicts had arisen between Ruotolo and Rold, his attorney in the civil rights action, and, in May, 2004, Rold withdrew as counsel for Ruotolo. Thereafter, M&N began representing Ruotolo in the civil rights action."

"Before the action could proceed to trial, the United States Supreme Court issued its decision in Garcetti v. Ceballus, 547 U.S. 410 (2006). In Garcetti, the Supreme Court held that the First Amendment does not “protect[] a government employee €from discipline based on speech made pursuant to the employee’s official duties.” 547 [J.S. at 41 3. Thereafter, the City defendants renewed their motion for summary judgment dismissing Ruotolo’s complaint on the grounds that the Report arose from Ruotolo’s official duties as a police officer, and thus was not protected by the First Amendment. In opposition, M&N argued that the amended complaint should in fairness be read to include Ruotolo’s conversation with the PBA attorney, which M&N contended were not pursuant to Ruotolo’s official job duties.
On July 19,2006, Judge Stein granted the City defendants’ motion for summary judgment, holding that Ruotolo prepared the Report pursuant to his official job duties and therefore had no First Amendment claim with respect to the Report. Ruotolo v. City of New York, 2006 U.S. Dist. LEXIS 49903,  (S.D.N.Y. July 19,2006).’ Because Ruotolo had not pled any claim based upon the conversation with the PBA attorney in either the initial or amended complaints, Judge Stein based his dismissal of the civil rights action solely on claims arising from the Report itself. However, Judge Stein noted that even if he had considered Ruotolo’s discussion with the PBA attorney, Ruotolo’s First Amendment claim would fail because that discussion was held pursuant to Ruotolo’s official job duties, which included ‘‘answering questions about safety issues at the
precinct.” Ruotolo v. City of New Yo& 2006 U.S. Dist. LEXIS 49903, “12-13 (S.D.N.Y.
July 19, 2006)"

"Here, defendants have made a prima face showing that Ruotolo would not have prevailed on his First Amendment claim even if defendants had amended the complaint to include Ruotolo’s conversations with the PBA attorney in April, 2000. The First Amendment does not protect public employees from adverse employment decisions based on communications they make pursuant to their official duties, see Weintraub v. Bd, of Educ., 593 F.3d 196, 200-01 (2d Cir. 2010), and defendants have submitted sufficient  evidence to show that Ruotolo’s conversation with the PRA attorney was made pursuant to his official duties."

"Defendants have also shown that Ruotolo would not have succeeded on a Due Process claim based on the City defendants’ deprivation of Ruotolo’s’s firearms. Where a government agency randomly and arbitrarily deprives a citizen of a protected property interest, due process is satisfied if the state provides an adequate post-deprivation remedy. Hellenic Am. Neighborhood Action Comm. v. New York City, 101 F.3d 877, 880 (2d Cir. 1996)."